1 Introduction
Given two topological vector spaces
$X,Y$
and a function
$c:X\times Y\rightarrow \mathbb{R}\cup \{-\infty ,+\infty \}$
, extending the Legendre–Fenchel conjugacy, Moreau [Reference Moreau22, Ch. 14, §3] defined, for any function
$g:Y\rightarrow \mathbb{R}\cup \{-\infty ,+\infty \}$
its
$c$
-conjugate as the function
$g^{c}:X\rightarrow \mathbb{R}\cup \{-\infty ,+\infty \}$
,
see §2 for the (extended) lower subtraction 
We refer to [Reference Balder4, Reference Dolecki6, Reference Dolecki and Kurcyusz7, Reference Elster and Wolf9, Reference Martinez-Legaz and Cambini17, Reference Moreau22, Reference Rubinov31, Reference Volle37] and the references therein for various duality results in such a context and for several applications. Given a function
$\unicode[STIX]{x1D711}:X\rightarrow \mathbb{R}\cup \{-\infty ,+\infty \}$
we will focus on the case
$c(x,y):=\unicode[STIX]{x1D711}(x-y)$
and
$Y=X$
. Otherwise stated, for a function
$f:X\rightarrow \mathbb{R}\cup \{-\infty ,+\infty \}$
we will be interested in the function
$f^{\unicode[STIX]{x1D711}}$
that we call the
$\unicode[STIX]{x1D711}$
-envelope of
$f$
, defined by
Our first aim in this paper is to study in great detail the structure of the transform
$f\mapsto f^{\unicode[STIX]{x1D711}}$
and provide various properties of
$\unicode[STIX]{x1D711}$
-envelopes.
On the other hand, considering the class
${\mathcal{B}}_{X}$
of closed balls of a Banach space
$X$
, Mazur [Reference Mazur19] studied some Banach spaces
$X$
for which every closed bounded convex subset is the intersection of some subclass of
${\mathcal{B}}_{X}$
; we refer to [Reference Granero, Jiménez-Sevilla and Moreno10] for a rich survey on the subject. Any such Banach space is actually called in the literature a Banach space with the Mazur intersection property. In his 1983 paper [Reference Vial36] Vial defined strongly convex sets of a normed space as convex sets which are intersections of closed balls with a common radius; sets which are intersections, for a fixed real
$r>0$
, of closed balls with radius equal to
$r$
are called
$r$
-strongly convex sets in [Reference Vial36]. This class of convex sets is thoroughly studied by Polovinkin [Reference Polovinkin25] (see also [Reference Polovinkin and Balashov26] and the references therein). Denoting by
$\mathbb{B}_{X}$
the closed unit ball of
$X$
centered at zero, any
$r$
-strongly convex set can be represented in the form
$$\begin{eqnarray}\mathop{\bigcap }_{x\in S}(x+r\mathbb{B}_{X})\quad \text{with some subset }S\subset X.\end{eqnarray}$$
So, given a subset
$\unicode[STIX]{x1D6EC}$
of the space
$X$
, our second aim in the paper is to analyze properties of the transform which assigns to each subset
$C$
of
$X$
the set
$$\begin{eqnarray}C^{\unicode[STIX]{x1D6EC}}:=\mathop{\bigcap }_{x\in C}(x+\unicode[STIX]{x1D6EC}).\end{eqnarray}$$
We will also provide the connections between the latter transform and the aforestated transform related to
$\unicode[STIX]{x1D711}$
-envelopes.
In §2 we recall the lower and upper additions (respectively subtractions), and we also recall various concepts and results in convex analysis which will be needed in the paper. Section 3 offers a large list of general properties of
$\unicode[STIX]{x1D711}$
-envelopes. Section 4 establishes the connections between
$\unicode[STIX]{x1D711}$
-envelopes and the aforementioned transform
$C\mapsto C^{\unicode[STIX]{x1D6EC}}$
; many properties of sets which can be represented in this form are also provided. In §5 we examine the question whether
$\unicode[STIX]{x1D713}=\unicode[STIX]{x1D711}(\cdot -a)-\unicode[STIX]{x1D6FC}$
(for some
$a\in X$
and
$\unicode[STIX]{x1D6FC}\in \mathbb{R}$
) whenever
$\unicode[STIX]{x1D713}$
is a
$\unicode[STIX]{x1D711}$
-envelope and
$\unicode[STIX]{x1D711}$
is a
$\unicode[STIX]{x1D713}$
-envelope. A counter-example is constructed and various sufficient conditions are given. The analogous question is also investigated with sets instead of functions. Section 6 considers additional properties in the case when the function
$\unicode[STIX]{x1D711}$
is either superadditive or subadditive. In §7, assuming that
$\unicode[STIX]{x1D711}$
is convex and lower semicontinuous, we provide several links between the
$\unicode[STIX]{x1D711}$
-envelope of a function and Legendre–Fenchel conjugates of other functions related to
$f$
. Taking
$\unicode[STIX]{x1D711}$
as a power of the norm, we also provide various results concerning the Klee envelope
$\unicode[STIX]{x1D705}_{\unicode[STIX]{x1D706},p}f$
(with index
$\unicode[STIX]{x1D706}$
and power
$p$
) of a function
$f$
, where
$$\begin{eqnarray}\unicode[STIX]{x1D705}_{\unicode[STIX]{x1D706},p}f(x):=\sup _{y\in X}\biggl(\frac{1}{p\unicode[STIX]{x1D706}}\Vert x-y\Vert ^{p}-f(y)\biggr)\quad \text{for all }x\in X.\end{eqnarray}$$
Finally in §8, assuming that
$-\unicode[STIX]{x1D711}$
is convex and lower semicontinuous, we continue to explore the links between
$\unicode[STIX]{x1D711}$
-envelopes and Legendre–Fenchel conjugates. By particularizing with
$\unicode[STIX]{x1D711}=-(1/p\unicode[STIX]{x1D706})\Vert \cdot \Vert ^{p}$
, for
$\unicode[STIX]{x1D706}>0$
and
$p\geqslant 1$
, we obtain several properties of Moreau envelopes with index
$\unicode[STIX]{x1D706}$
and power
$p$
.
2 Preliminaries
Following Moreau [Reference Moreau22], we extend the usual addition on
$\mathbb{R}$
to
$\overline{\mathbb{R}}=[-\infty ,+\infty ]$
. We define the upper addition 
and the lower addition 
as the laws extending the usual addition via the following conventions:
This leads to the introduction of the upper subtraction 
and the lower subtraction 
respectively defined by
Let
$X$
be a vector space; all vector spaces will be real vector spaces. Given two extended real-valued functions
$f$
,
$g:X\rightarrow \overline{\mathbb{R}}$
, the (Moreau) inf-convolution (also called infimal convolution) of
$f$
and
$g$
is defined as follows: for every
$x\in X$
,
In a symmetric way, the (Moreau) sup-convolution (or supremal convolution) of
$f$
and
$g$
is defined by
For the function
$f$
as above, the set
$\text{dom}\,f=\{x\in X,f(x)<+\infty \}$
is called the effective domain of
$f$
. We call
$f$
a proper function if
$f(x)<+\infty$
for at least one
$x\in X$
, and
$f(x)>-\infty$
for all
$x\in X$
, or in other words, if
$\text{dom}\,f$
is a non-empty set on which
$f$
is finite. The function which is constantly equal to
$+\infty$
(respectively
$-\infty$
) on
$X$
is denoted by
$\unicode[STIX]{x1D714}_{X}$
(respectively
$-\unicode[STIX]{x1D714}_{X}$
).
Now assume that
$X$
is a locally convex space; all such spaces in the paper will be Hausdorff. We will denote by
$X^{\ast }$
the topological dual of
$X$
. Then, following again [Reference Moreau22], we set
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6E4}(X) & := & \displaystyle \{\!f:X\rightarrow \overline{\mathbb{R}},f\text{ is a pointwise supremum of a family of continuous}\nonumber\\ \displaystyle & & \displaystyle \text{affine functions with slopes in }X^{\ast }\!\}\nonumber\end{eqnarray}$$
and
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6E4}(X^{\ast }) & := & \displaystyle \{\!g:X^{\ast }\rightarrow \overline{\mathbb{R}},g\text{ is a pointwise supremum of a family of continuous}\nonumber\\ \displaystyle & & \displaystyle \text{affine functions with slopes in }X\!\}.\nonumber\end{eqnarray}$$
We denote by
$\unicode[STIX]{x1D6E4}_{0}(X)$
the set of
$f\in \unicode[STIX]{x1D6E4}(X)$
which differ from
$\unicode[STIX]{x1D714}_{X}$
and
$-\unicode[STIX]{x1D714}_{X}$
. In the same way,
$\unicode[STIX]{x1D6E4}_{0}(X^{\ast })$
is the set
$\unicode[STIX]{x1D6E4}_{0}(X^{\ast })=\unicode[STIX]{x1D6E4}(X^{\ast })\setminus \{\unicode[STIX]{x1D714}_{X^{\ast }},-\unicode[STIX]{x1D714}_{X^{\ast }}\}$
. The classes
$\unicode[STIX]{x1D6E4}_{0}(X)$
and
$\unicode[STIX]{x1D6E4}_{0}(X^{\ast })$
are respectively characterized by
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6E4}_{0}(X) & = & \displaystyle \{f:X\rightarrow \overline{\mathbb{R}},f\text{ is closed, convex and proper}\}\nonumber\\ \displaystyle & = & \displaystyle \{f:X\rightarrow \overline{\mathbb{R}},f\text{ is }w(X,X^{\ast })\text{ closed, convex and proper}\},\nonumber\end{eqnarray}$$
and
$$\begin{eqnarray}\unicode[STIX]{x1D6E4}_{0}(X^{\ast })=\{g:X^{\ast }\rightarrow \overline{\mathbb{R}},g\text{ is }w(X^{\ast },X)\text{ closed, convex and proper}\},\end{eqnarray}$$
see, for example, [Reference Attouch, Buttazzo and Michaille1, Reference Ekeland and Temam8, Reference Moreau22]. Above and in the rest of the paper,
$w(X,X^{\ast })$
and
$w(X^{\ast },X)$
stand for the weak topology on
$X$
and the weak star topology on
$X^{\ast }$
, respectively.
With the function
$f:X\rightarrow \overline{\mathbb{R}}$
is associated, in the duality pairing from
$X$
to
$X^{\ast }$
, its Legendre–Fenchel conjugate
$f^{\ast }:X^{\ast }\rightarrow \overline{\mathbb{R}}$
defined by
$$\begin{eqnarray}\text{for all }x^{\ast }\in X^{\ast },\quad f^{\ast }(x^{\ast })=\sup _{\unicode[STIX]{x1D709}\in X}\{\langle x^{\ast },\unicode[STIX]{x1D709}\rangle -f(\unicode[STIX]{x1D709})\}.\end{eqnarray}$$
In the same way, throughout the paper (unless otherwise stated) the Legendre–Fenchel conjugate of a function
$g:X^{\ast }\rightarrow \overline{\mathbb{R}}$
defined on the dual space
$X^{\ast }$
will be taken in the duality pairing from
$X^{\ast }$
to
$X$
, that is,
$g^{\ast }:X\rightarrow \overline{\mathbb{R}}$
is defined on
$X$
by
$$\begin{eqnarray}\text{for all }x\in X,\quad g^{\ast }(x)=\sup _{\unicode[STIX]{x1D709}^{\ast }\in X^{\ast }}\{\langle \unicode[STIX]{x1D709}^{\ast },x\rangle -g(\unicode[STIX]{x1D709}^{\ast })\}.\end{eqnarray}$$
The Legendre–Fenchel transform
$f\mapsto f^{\ast }$
(see, for example, [Reference Moreau22]) is known to be a one-to-one mapping from
$\unicode[STIX]{x1D6E4}_{0}(X)$
onto
$\unicode[STIX]{x1D6E4}_{0}(X^{\ast })$
. For any
$f\in \unicode[STIX]{x1D6E4}_{0}(X)$
one has
$f=f^{\ast \ast }$
and for any
$g\in \unicode[STIX]{x1D6E4}_{0}(X^{\ast })$
one has
$g=g^{\ast \ast }$
(see, for example, [Reference Attouch, Buttazzo and Michaille1, Reference Ekeland and Temam8, Reference Moreau22]).
Given a set
$C\subset X$
, we denote as usual by
$\unicode[STIX]{x1D6FF}_{C}$
the indicator function of
$C$
, i.e.
$\unicode[STIX]{x1D6FF}_{C}(x)=0$
if
$x\in C$
and
$\unicode[STIX]{x1D6FF}_{C}(x)=+\infty$
if
$x\notin C$
. The support function
$\unicode[STIX]{x1D70E}_{C}:X^{\ast }\rightarrow \overline{\mathbb{R}}$
of
$C$
is defined by
$$\begin{eqnarray}\text{for all }x^{\ast }\in X^{\ast },\quad \unicode[STIX]{x1D70E}_{C}(x^{\ast })=\sup _{\unicode[STIX]{x1D709}\in C}\langle x^{\ast },\unicode[STIX]{x1D709}\rangle ,\end{eqnarray}$$
so
$\unicode[STIX]{x1D70E}_{C}$
coincides with the Legendre–Fenchel conjugate of
$\unicode[STIX]{x1D6FF}_{C}$
. For a non-empty cone
$K\subset X$
, the support function
$\unicode[STIX]{x1D70E}_{K}$
is equal to the indicator function of the polar cone
$K^{\circ }$
of
$K$
defined by
$$\begin{eqnarray}K^{\circ }=\{x^{\ast }\in X^{\ast },\langle x^{\ast },x\rangle \leqslant 0\text{ for all }x\in K\}.\end{eqnarray}$$
For a set
$C\subset X$
, we denote by
$\text{co}(C)$
(respectively
$\overline{\text{co}}(C)$
) the convex hull (respectively closed convex hull) of
$C$
. The
$w(X^{\ast },X)$
-closed convex hull of a set
$D\subset X^{\ast }$
is denoted by
$\overline{\text{co}}^{w\ast }(D)$
. For a function
$f:X\rightarrow \overline{\mathbb{R}}$
, its convex hull
$\text{co}(f)$
(respectively lower semicontinuous convex hull
$\overline{\text{co}}(f)$
) is the greatest convex (respectively lower semicontinuous convex) function less than or equal to
$f$
. The
$w(X^{\ast },X)$
-lower semicontinuous convex hull of a function
$g:X^{\ast }\rightarrow \overline{\mathbb{R}}$
is denoted by
$\overline{\text{co}}^{w\ast }(g)$
.
If
$f\in \unicode[STIX]{x1D6E4}_{0}(X)$
and if
$\overline{x}\in \text{dom}\,f$
, the recession function
$f^{\infty }$
is defined by
$$\begin{eqnarray}\text{for all }u\in X,\quad f^{\infty }(u)=\lim _{t\rightarrow +\infty }\frac{f(\overline{x}+tu)-f(\overline{x})}{t}=\sup _{t>0}\frac{f(\overline{x}+tu)-f(\overline{x})}{t}.\end{eqnarray}$$
The function
$f^{\infty }:X\rightarrow \mathbb{R}\cup \{+\infty \}$
does not depend on the point
$\overline{x}\in \text{dom}\,f$
since it is also given by
$$\begin{eqnarray}\text{for all }u\in X,\quad f^{\infty }(u)=\sup _{x\in \text{dom}\,f}(f(x+u)-f(x)).\end{eqnarray}$$
The function
$f^{\infty }$
satisfies
$f^{\infty }\in \unicode[STIX]{x1D6E4}_{0}(X)$
, it is positively homogeneous and we have
$f^{\infty }=\unicode[STIX]{x1D70E}_{\text{dom}\,f^{\ast }}$
. Given a closed convex set
$C\subset X$
and
$\overline{x}\in C$
, the recession cone
$C^{\infty }$
is defined by
$$\begin{eqnarray}C^{\infty }=\{u\in X,\overline{x}+tu\in C\text{ for all }t\geqslant 0\}.\end{eqnarray}$$
The set
$C^{\infty }$
does not depend on
$\overline{x}\in C$
and is also given by
$$\begin{eqnarray}C^{\infty }=\{u\in X,u+C\subset C\}.\end{eqnarray}$$
It follows from the definition that
$C^{\infty }$
is a closed convex cone and we have
$\unicode[STIX]{x1D6FF}_{C^{\infty }}=~(\unicode[STIX]{x1D6FF}_{C})^{\infty }$
. For more details on recession analysis, see, for example, [Reference Attouch, Buttazzo and Michaille1, Reference Auslender and Teboulle2, Reference Hiriart-Urruty and Lemaréchal13, Reference Rockafellar28].
Let us end these preliminaries with the subdifferential of convex analysis. We recall that the subdifferential
$\unicode[STIX]{x2202}f(x)$
of a convex function
$f:X\rightarrow \mathbb{R}\cup \{+\infty \}$
at
$x\in \text{dom}\,f$
is the set
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x2202}f(x)=\{\unicode[STIX]{x1D709}^{\ast }\in X^{\ast },f(y)\geqslant f(x)+\langle \unicode[STIX]{x1D709}^{\ast },y-x\rangle \text{for every }y\in X\}. & & \displaystyle\end{eqnarray}$$
When
$x\notin \text{dom}\,f$
, then
$\unicode[STIX]{x2202}f(x)=\emptyset$
by convention. The domain and the range of the operator 
are respectively given by
$$\begin{eqnarray}\displaystyle \text{dom}(\unicode[STIX]{x2202}f) & = & \displaystyle \{x\in X,\unicode[STIX]{x2202}f(x)\neq \emptyset \},\nonumber\\ \displaystyle \text{Rge}(\unicode[STIX]{x2202}f) & = & \displaystyle \{x^{\ast }\in X^{\ast },\exists x\in X,x^{\ast }\in \unicode[STIX]{x2202}f(x)\}.\nonumber\end{eqnarray}$$
If
$f\in \unicode[STIX]{x1D6E4}_{0}(X)$
, the subdifferentials of
$f$
and
$f^{\ast }$
are connected through the following relation
$$\begin{eqnarray}\displaystyle x^{\ast }\in \unicode[STIX]{x2202}f(x)\;\Longleftrightarrow \;x\in \unicode[STIX]{x2202}f^{\ast }(x^{\ast }), & & \displaystyle\end{eqnarray}$$
for all
$x\in X$
and
$x^{\ast }\in X^{\ast }$
. For further details, the reader is referred to the classical textbooks on convex analysis, see, for example, [Reference Hiriart-Urruty and Lemaréchal13, Reference Rockafellar28].
3 Definitions, general properties
Let
$X$
be a vector space. For functions
$\unicode[STIX]{x1D711}:X\rightarrow \overline{\mathbb{R}}$
and
$f:X\rightarrow \overline{\mathbb{R}}$
, the
$\unicode[STIX]{x1D711}$
-envelope of
$f$
is defined as follows:
A function
$g:X\rightarrow \overline{\mathbb{R}}$
is said to be a
$\unicode[STIX]{x1D711}$
-envelope if there exists
$f:X\rightarrow \overline{\mathbb{R}}$
such that
$g=f^{\unicode[STIX]{x1D711}}$
. It is immediate to check that for every function
$f:X\rightarrow \overline{\mathbb{R}}$
,
$$\begin{eqnarray}f^{-\unicode[STIX]{x1D714}_{X}}=-\unicode[STIX]{x1D714}_{X},\quad \text{while }f^{\unicode[STIX]{x1D714}_{X}}=\left\{\begin{array}{@{}ll@{}}\displaystyle \unicode[STIX]{x1D714}_{X} & \text{if }f\neq \unicode[STIX]{x1D714}_{X},\\ \displaystyle -\unicode[STIX]{x1D714}_{X} & \text{if }f=\unicode[STIX]{x1D714}_{X}.\end{array}\right.\end{eqnarray}$$
It ensues that the unique
$(-\unicode[STIX]{x1D714}_{X})$
-envelope is the function
$-\unicode[STIX]{x1D714}_{X}$
, while the
$\unicode[STIX]{x1D714}_{X}$
-envelopes are
$\pm \unicode[STIX]{x1D714}_{X}$
. The function
$f^{\unicode[STIX]{x1D711}}$
can be expressed via the inf-convolution and sup-convolution operators
$$\begin{eqnarray}\displaystyle f^{\unicode[STIX]{x1D711}}=\unicode[STIX]{x1D711}\triangle (-f)=-((-\unicode[STIX]{x1D711}){\bigtriangledown}f). & & \displaystyle\end{eqnarray}$$
The roles played by
$f$
and
$\unicode[STIX]{x1D711}$
in the definition of
$f^{\unicode[STIX]{x1D711}}$
are opposite in the sense that
$$\begin{eqnarray}\displaystyle (-\unicode[STIX]{x1D711})^{(-f)}=(-f)\triangle (-(-\unicode[STIX]{x1D711}))=(-f)\triangle \unicode[STIX]{x1D711}=f^{\unicode[STIX]{x1D711}}. & & \displaystyle\end{eqnarray}$$
The definition of
$f^{\unicode[STIX]{x1D711}}$
is closely connected to the deconvolution operation. For any
$g$
,
$h:X\rightarrow \overline{\mathbb{R}}$
, the deconvolution of
$g$
and
$h$
is the function
$g\ominus h$
defined by
for every
$x\in X$
. Denoting by
$h_{-}$
the function defined by
$h_{-}(x)=h(-x)$
for every
$x\in X$
, we deduce immediately from the above definition that
$$\begin{eqnarray}\displaystyle g\ominus h=g\triangle (-h_{-})=(h_{-})^{g}. & & \displaystyle\end{eqnarray}$$
It ensues that for any
$f$
,
$\unicode[STIX]{x1D711}:X\rightarrow \overline{\mathbb{R}}$
,
$$\begin{eqnarray}f^{\unicode[STIX]{x1D711}}=\unicode[STIX]{x1D711}\ominus f_{-}.\end{eqnarray}$$
The deconvolution operation has been studied in detail by many authors, see, for example, [Reference Aze and Volle3, Reference Hiriart-Urruty12, Reference Hiriart-Urruty and Mazure14, Reference Volle38].
Following the terminology of Moreau [Reference Moreau21], we call
$\unicode[STIX]{x1D711}$
-elementary function a function of the form
$\unicode[STIX]{x1D711}(\cdot -y)+\unicode[STIX]{x1D706}$
with
$y\in X$
and
$\unicode[STIX]{x1D706}\in \mathbb{R}$
. By using a generalized conjugacy argument, one can show that for any
$\unicode[STIX]{x1D711}$
,
$f:X\rightarrow \overline{\mathbb{R}}$
,
$$\begin{eqnarray}\displaystyle (f^{\unicode[STIX]{x1D711}_{-}})^{\unicode[STIX]{x1D711}}\text{ is the upper envelope of the }\unicode[STIX]{x1D711}\text{-elementary functions that minorize }f, & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$
see, for example, [Reference Moreau21, §4] and [Reference Rockafellar and Wets30, §11.L]. One can easily deduce the following characterization of
$\unicode[STIX]{x1D711}$
-envelopes: for any
$g:X\rightarrow \overline{\mathbb{R}}$
,
$$\begin{eqnarray}\displaystyle & \displaystyle g\text{ is the upper envelope of a family of }\unicode[STIX]{x1D711}\text{-elementary functions} & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle \Updownarrow & \displaystyle \nonumber\\ \displaystyle & \displaystyle g=(g^{\unicode[STIX]{x1D711}_{-}})^{\unicode[STIX]{x1D711}} & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle \Updownarrow & \displaystyle \nonumber\\ \displaystyle & \displaystyle g\text{ is a }\unicode[STIX]{x1D711}\text{-envelope}. & \displaystyle\end{eqnarray}$$
The expression of the double envelope
$(g^{\unicode[STIX]{x1D711}_{-}})^{\unicode[STIX]{x1D711}}$
can be developed as follows
$$\begin{eqnarray}\displaystyle (g^{\unicode[STIX]{x1D711}_{-}})^{\unicode[STIX]{x1D711}} & = & \displaystyle \unicode[STIX]{x1D711}\triangle (-g^{\unicode[STIX]{x1D711}_{-}})\nonumber\\ \displaystyle & = & \displaystyle \unicode[STIX]{x1D711}\triangle (-(\unicode[STIX]{x1D711}_{-}\triangle (-g)))\nonumber\\ \displaystyle & = & \displaystyle \unicode[STIX]{x1D711}\triangle ((-\unicode[STIX]{x1D711}_{-}){\bigtriangledown}g).\nonumber\end{eqnarray}$$
By using the deconvolution operation, we obtain
$$\begin{eqnarray}\displaystyle (g^{\unicode[STIX]{x1D711}_{-}})^{\unicode[STIX]{x1D711}} & = & \displaystyle \unicode[STIX]{x1D711}\ominus (\unicode[STIX]{x1D711}\triangle (-g_{-}))\nonumber\\ \displaystyle & = & \displaystyle \unicode[STIX]{x1D711}\ominus (\unicode[STIX]{x1D711}\ominus g).\nonumber\end{eqnarray}$$
From the equivalence (8)
$\Leftrightarrow$
(9), we deduce that
$$\begin{eqnarray}\displaystyle \begin{array}{@{}c@{}}\displaystyle g\text{ is a }\unicode[STIX]{x1D711}\text{-envelope}\\ \displaystyle \Updownarrow \\ \displaystyle g=\unicode[STIX]{x1D711}\triangle ((-\unicode[STIX]{x1D711}_{-}){\bigtriangledown}g)\\ \displaystyle \Updownarrow \\ \displaystyle g=\unicode[STIX]{x1D711}\ominus (\unicode[STIX]{x1D711}\ominus g).\end{array} & & \displaystyle\end{eqnarray}$$
Now let
$f$
,
$\unicode[STIX]{x1D713}:X\rightarrow \overline{\mathbb{R}}$
. Following the terminology of Martinez-Legaz and Penot [Reference Martinez-Legaz and Penot18], the function
$f$
is said to be (exactly)
$\unicode[STIX]{x1D713}$
-regular if
$f=(f\ominus \unicode[STIX]{x1D713}){\bigtriangledown}\unicode[STIX]{x1D713}$
. By taking the opposite in each member of the equality (10), we find
$$\begin{eqnarray}\displaystyle -g & = & \displaystyle (-\unicode[STIX]{x1D711}){\bigtriangledown}(\unicode[STIX]{x1D711}_{-}\triangle (-g))\nonumber\\ \displaystyle & = & \displaystyle (-\unicode[STIX]{x1D711}){\bigtriangledown}((-g)\ominus (-\unicode[STIX]{x1D711})).\nonumber\end{eqnarray}$$
In view of the above equivalences, this implies that
$$\begin{eqnarray}g\text{ is a }\unicode[STIX]{x1D711}\text{-envelope}\;\Longleftrightarrow \;-g\text{ is }(-\unicode[STIX]{x1D711})\text{-regular in the sense of [18]}.\end{eqnarray}$$
We denote by
${\mathcal{E}}^{\unicode[STIX]{x1D711}}(X)$
, or
${\mathcal{E}}^{\unicode[STIX]{x1D711}}$
if there is no risk of confusion, the set of
$\unicode[STIX]{x1D711}$
-envelopes and by
$F_{\unicode[STIX]{x1D711}}:{\mathcal{E}}^{\unicode[STIX]{x1D711}_{-}}\rightarrow {\mathcal{E}}^{\unicode[STIX]{x1D711}}$
the map defined by
$F_{\unicode[STIX]{x1D711}}(f)=f^{\unicode[STIX]{x1D711}}$
for every
$f\in {\mathcal{E}}^{\unicode[STIX]{x1D711}_{-}}$
. The equivalence (8)
$\Leftrightarrow$
(9) says that
$F_{\unicode[STIX]{x1D711}}\circ F_{\unicode[STIX]{x1D711}_{-}}=\text{Id}_{{\mathcal{E}}^{\unicode[STIX]{x1D711}}}$
and
$F_{\unicode[STIX]{x1D711}_{-}}\circ F_{\unicode[STIX]{x1D711}}=\text{Id}_{{\mathcal{E}}^{\unicode[STIX]{x1D711}_{-}}}$
, otherwise stated we have the following proposition.
Proposition 3.1. The map
$F_{\unicode[STIX]{x1D711}}:{\mathcal{E}}^{\unicode[STIX]{x1D711}_{-}}\rightarrow {\mathcal{E}}^{\unicode[STIX]{x1D711}}$
is bijective and
$(F_{\unicode[STIX]{x1D711}})^{-1}=F_{\unicode[STIX]{x1D711}_{-}}$
.
As a consequence of the previous proposition, if
$\unicode[STIX]{x1D711}$
is even the map
$F_{\unicode[STIX]{x1D711}}:{\mathcal{E}}^{\unicode[STIX]{x1D711}}\rightarrow {\mathcal{E}}^{\unicode[STIX]{x1D711}}$
is bijective and
$(F_{\unicode[STIX]{x1D711}})^{-1}=F_{\unicode[STIX]{x1D711}}$
.
Let us now state several general properties of
$\unicode[STIX]{x1D711}$
-envelopes.
Proposition 3.2. Let
$X$
be a vector space and let
$\unicode[STIX]{x1D711}:X\rightarrow \overline{\mathbb{R}}$
.
-
(i) For every function
$f:X\rightarrow \overline{\mathbb{R}}$
and every
$a\in X$
and
$\unicode[STIX]{x1D6FD}\in \mathbb{R}$
, we have
$(f(\cdot -a)-\unicode[STIX]{x1D6FD})^{\unicode[STIX]{x1D711}}=f^{\unicode[STIX]{x1D711}}(\cdot -a)+\unicode[STIX]{x1D6FD}$
. If
$g\in {\mathcal{E}}^{\unicode[STIX]{x1D711}}$
, then
$g(\cdot -a)+\unicode[STIX]{x1D6FD}\in {\mathcal{E}}^{\unicode[STIX]{x1D711}}$
for every
$a\in X$
and
$\unicode[STIX]{x1D6FD}\in \mathbb{R}$
. -
(ii) Given a family
$(f_{i})_{i\in I}$
of functions
$f_{i}:X\rightarrow \overline{\mathbb{R}}$
, we have
$(\inf _{i\in I}f_{i})^{\unicode[STIX]{x1D711}}=\sup _{i\in I}f_{i}^{\unicode[STIX]{x1D711}}$
. If
$g=\sup _{i\in I}g_{i}$
with
$g_{i}\in {\mathcal{E}}^{\unicode[STIX]{x1D711}}$
for every
$i\in I$
, then
$g\in {\mathcal{E}}^{\unicode[STIX]{x1D711}}$
. -
(iii) For
$f_{1}$
,
$f_{2}:X\rightarrow \overline{\mathbb{R}}$
, we have
$(f_{1}{\bigtriangledown}f_{2})^{\unicode[STIX]{x1D711}}=f_{1}^{(f_{2}^{\unicode[STIX]{x1D711}})}$
. Let
$g$
,
$h:X\rightarrow \overline{\mathbb{R}}$
. If
$h\in {\mathcal{E}}^{g}$
and
$g\in {\mathcal{E}}^{\unicode[STIX]{x1D711}}$
, then
$h\in {\mathcal{E}}^{\unicode[STIX]{x1D711}}$
. Otherwise stated, if
$g\in {\mathcal{E}}^{\unicode[STIX]{x1D711}}$
, then
${\mathcal{E}}^{g}\subset {\mathcal{E}}^{\unicode[STIX]{x1D711}}$
. -
(iv) For
$f:X\rightarrow \overline{\mathbb{R}}$
, we have
$(f^{\unicode[STIX]{x1D711}})_{-}={f_{-}}^{\unicode[STIX]{x1D711}_{-}}$
. As a consequence,
$g\in {\mathcal{E}}^{\unicode[STIX]{x1D711}}$
if and only if
$g_{-}\in {\mathcal{E}}^{\unicode[STIX]{x1D711}_{-}}$
.
Proof. (i) Let
$a\in X$
and
$\unicode[STIX]{x1D6FD}\in \mathbb{R}$
. For every
$x\in X$
, we have
For the second assertion of (i), it suffices to apply the first part with
$g=f^{\unicode[STIX]{x1D711}}$
.
(ii) By definition, we have
$$\begin{eqnarray}\displaystyle \Bigl(\inf _{i\in I}f_{i}\Bigr)^{\unicode[STIX]{x1D711}} & = & \displaystyle \unicode[STIX]{x1D711}\triangle \Bigl(-\!\inf _{i\in I}f_{i}\Bigr)\nonumber\\ \displaystyle & = & \displaystyle \unicode[STIX]{x1D711}\triangle \sup _{i\in I}(-f_{i})\nonumber\\ \displaystyle & = & \displaystyle \sup _{i\in I}(\unicode[STIX]{x1D711}\triangle (-f_{i}))=\sup _{i\in I}f_{i}^{\unicode[STIX]{x1D711}},\quad \text{see, for example, [21]}.\nonumber\end{eqnarray}$$
Now assume that
$g=\sup _{i\in I}g_{i}$
with
$g_{i}\in {\mathcal{E}}^{\unicode[STIX]{x1D711}}$
for every
$i\in I$
. Then, for each
$i\in I$
, we have
$g_{i}=f_{i}^{\unicode[STIX]{x1D711}}$
for some
$f_{i}$
. It ensues that
$g=\sup _{i\in I}f_{i}^{\unicode[STIX]{x1D711}}=(\inf _{i\in I}f_{i})^{\unicode[STIX]{x1D711}}$
, hence
$g\in {\mathcal{E}}^{\unicode[STIX]{x1D711}}$
.
(iii) By definition, we have
$$\begin{eqnarray}\displaystyle f_{1}^{(f_{2}^{\unicode[STIX]{x1D711}})} & = & \displaystyle f_{2}^{\unicode[STIX]{x1D711}}\triangle (-f_{1})\nonumber\\ \displaystyle & = & \displaystyle (\unicode[STIX]{x1D711}\triangle (-f_{2}))\triangle (-f_{1})\nonumber\\ \displaystyle & = & \displaystyle \unicode[STIX]{x1D711}\triangle ((-f_{2})\triangle (-f_{1}))\nonumber\\ \displaystyle & = & \displaystyle \unicode[STIX]{x1D711}\triangle (-(f_{2}{\bigtriangledown}f_{1}))\nonumber\\ \displaystyle & = & \displaystyle (f_{2}{\bigtriangledown}f_{1})^{\unicode[STIX]{x1D711}}=(f_{1}{\bigtriangledown}f_{2})^{\unicode[STIX]{x1D711}}.\nonumber\end{eqnarray}$$
Now assume that
$h\in {\mathcal{E}}^{g}$
and
$g\in {\mathcal{E}}^{\unicode[STIX]{x1D711}}$
. Then there exist
$f_{1}$
,
$f_{2}:X\rightarrow \overline{\mathbb{R}}$
such that
$h=f_{1}^{g}$
and
$g=f_{2}^{\unicode[STIX]{x1D711}}$
. It ensues that
$h=f_{1}^{(f_{2}^{\unicode[STIX]{x1D711}})}=(f_{1}{\bigtriangledown}f_{2})^{\unicode[STIX]{x1D711}}$
, hence
$h\in {\mathcal{E}}^{\unicode[STIX]{x1D711}}$
.
(iv) For every
$x\in X$
, we have
If
$g\in {\mathcal{E}}^{\unicode[STIX]{x1D711}}$
, there exists
$f:X\rightarrow \overline{\mathbb{R}}$
such that
$g=f^{\unicode[STIX]{x1D711}}$
. It ensues that
$g_{-}=(f^{\unicode[STIX]{x1D711}})_{-}=(f_{-})^{\unicode[STIX]{x1D711}_{-}}$
, hence
$g_{-}\in {\mathcal{E}}^{\unicode[STIX]{x1D711}_{-}}$
. The proof of the reverse assertion is identical.◻
The equalities in assertions (i) and (ii) of the above proposition are also consequences of [Reference Singer, Hammer and Pallaschke32, Theorem 3.1]Footnote
1
giving a general description of generalized conjugacy from
${\mathcal{F}}(X,\overline{\mathbb{R}})$
(the set of functions from
$X$
to
$\overline{\mathbb{R}}$
) into itself. The above proofs are provided for completeness.
In the next proposition, we show that the
$\unicode[STIX]{x1D711}$
-envelope of a continuous linear functional is affine and we characterize the elements of
${\mathcal{E}}^{\unicode[STIX]{x1D711}}$
that are linear.
Proposition 3.3. Let
$X$
be a locally convex space. Let
$\unicode[STIX]{x1D711}:X\rightarrow \overline{\mathbb{R}}$
and
$\unicode[STIX]{x1D709}^{\ast }\in X^{\ast }$
. Then we have:
-
(i)
$\langle \unicode[STIX]{x1D709}^{\ast },\cdot \rangle ^{\unicode[STIX]{x1D711}}=-\langle \unicode[STIX]{x1D709}^{\ast },\cdot \rangle +(-\unicode[STIX]{x1D711})^{\ast }(\unicode[STIX]{x1D709}^{\ast })$
; -
(ii) if
$\unicode[STIX]{x1D711}\neq -\unicode[STIX]{x1D714}_{X}$
, the following equivalence holds
$$\begin{eqnarray}\langle \unicode[STIX]{x1D709}^{\ast },\cdot \rangle \in {\mathcal{E}}^{\unicode[STIX]{x1D711}}\;\Longleftrightarrow \;\unicode[STIX]{x1D709}^{\ast }\in -\text{dom}(-\unicode[STIX]{x1D711})^{\ast }.\end{eqnarray}$$
Proof. (i) For every
$x\in X$
, we have
$$\begin{eqnarray}\displaystyle \langle \unicode[STIX]{x1D709}^{\ast },\cdot \rangle ^{\unicode[STIX]{x1D711}}(x) & = & \displaystyle \sup _{y\in X}\{\unicode[STIX]{x1D711}(y)-\langle \unicode[STIX]{x1D709}^{\ast },x-y\rangle \}\nonumber\\ \displaystyle & = & \displaystyle -\langle \unicode[STIX]{x1D709}^{\ast },x\rangle +(-\unicode[STIX]{x1D711})^{\ast }(\unicode[STIX]{x1D709}^{\ast }).\nonumber\end{eqnarray}$$
(ii) Let
$g=\langle \unicode[STIX]{x1D709}^{\ast },\cdot \rangle$
. We deduce from (i) that
$$\begin{eqnarray}\displaystyle g^{\unicode[STIX]{x1D711}_{-}}=-\langle \unicode[STIX]{x1D709}^{\ast },\cdot \rangle +(-\unicode[STIX]{x1D711}_{-})^{\ast }(\unicode[STIX]{x1D709}^{\ast })=-\langle \unicode[STIX]{x1D709}^{\ast },\cdot \rangle +(-\unicode[STIX]{x1D711})^{\ast }(-\unicode[STIX]{x1D709}^{\ast }). & & \displaystyle\end{eqnarray}$$
First assume that
$(-\unicode[STIX]{x1D711})^{\ast }(-\unicode[STIX]{x1D709}^{\ast })=+\infty$
. Then we have
$g^{\unicode[STIX]{x1D711}_{-}}=\unicode[STIX]{x1D714}_{X}$
, thus implying that
$(g^{\unicode[STIX]{x1D711}_{-}})^{\unicode[STIX]{x1D711}}=-\unicode[STIX]{x1D714}_{X}$
. It ensues that
$(g^{\unicode[STIX]{x1D711}_{-}})^{\unicode[STIX]{x1D711}}\neq g$
, which shows that
$g\notin {\mathcal{E}}^{\unicode[STIX]{x1D711}}$
according to the equivalence (7)
$\Longleftrightarrow$
(8). Now assume that
$(-\unicode[STIX]{x1D711})^{\ast }(-\unicode[STIX]{x1D709}^{\ast })<+\infty$
. Observe that
$(-\unicode[STIX]{x1D711})^{\ast }(-\unicode[STIX]{x1D709}^{\ast })\in \mathbb{R}$
since
$$\begin{eqnarray}(-\unicode[STIX]{x1D711})^{\ast }(-\unicode[STIX]{x1D709}^{\ast })=-\infty \;\Longrightarrow \;\sup _{x\in X}\langle -\unicode[STIX]{x1D709}^{\ast },x\rangle +\unicode[STIX]{x1D711}(x)=-\infty \;\Longrightarrow \;\unicode[STIX]{x1D711}=-\unicode[STIX]{x1D714}_{X},\end{eqnarray}$$
which is impossible by assumption. Since
$(-\unicode[STIX]{x1D711})^{\ast }(-\unicode[STIX]{x1D709}^{\ast })\in \mathbb{R}$
, we deduce from (11), (i) above and Proposition 3.2(i) that
$$\begin{eqnarray}(g^{\unicode[STIX]{x1D711}_{-}})^{\unicode[STIX]{x1D711}}=\langle \unicode[STIX]{x1D709}^{\ast },\cdot \rangle +(-\unicode[STIX]{x1D711})^{\ast }(-\unicode[STIX]{x1D709}^{\ast })-(-\unicode[STIX]{x1D711})^{\ast }(-\unicode[STIX]{x1D709}^{\ast })=\langle \unicode[STIX]{x1D709}^{\ast },\cdot \rangle =g,\end{eqnarray}$$
and therefore
$g\in {\mathcal{E}}^{\unicode[STIX]{x1D711}}$
.◻
For every set
$C\subset X$
, let us set
$$\begin{eqnarray}\unicode[STIX]{x1D6F4}_{C}=\{f:X\rightarrow \overline{\mathbb{R}},\text{dom}\,f\subset C\}\qquad \text{and}\qquad \unicode[STIX]{x1D6F4}_{C}^{\ast }=\{f^{\ast },f\in \unicode[STIX]{x1D6F4}_{C}\}.\end{eqnarray}$$
We adopt the same notations
$\unicode[STIX]{x1D6F4}_{D}$
and
$\unicode[STIX]{x1D6F4}_{D}^{\ast }$
for a subset
$D\subset X^{\ast }$
.
Theorem 3.1. Let
$X$
be a locally convex space and let
$\unicode[STIX]{x1D711}:X\rightarrow \overline{\mathbb{R}}$
be such that
$\unicode[STIX]{x1D711}\neq -\unicode[STIX]{x1D714}_{X}$
. For every subset
$D$
of
$X^{\ast }$
, the following assertions are equivalent:
-
(i)
$\unicode[STIX]{x1D6F4}_{D}^{\ast }\subset {\mathcal{E}}^{\unicode[STIX]{x1D711}}$
; -
(ii)
$\{f\in \unicode[STIX]{x1D6E4}_{0}(X),\text{dom}\,f^{\ast }\subset D\}\subset {\mathcal{E}}^{\unicode[STIX]{x1D711}}$
; -
(iii)
$D\subset -\text{dom}(-\unicode[STIX]{x1D711})^{\ast }$
.
Proof. (i)
$\Rightarrow$
(ii) Let
$D\subset X^{\ast }$
. Observe that
$$\begin{eqnarray}\displaystyle \{f\in \unicode[STIX]{x1D6E4}_{0}(X),\text{dom}\,f^{\ast }\subset D\} & = & \displaystyle \{g^{\ast },\text{dom}\,g\subset D\text{ and }g\in \unicode[STIX]{x1D6E4}_{0}(X)\}\nonumber\\ \displaystyle & \subset & \displaystyle \{g^{\ast },\text{dom}\,g\subset D\}=\unicode[STIX]{x1D6F4}_{D}^{\ast }.\nonumber\end{eqnarray}$$
The implication (i)
$\Rightarrow$
(ii) follows immediately.
(ii)
$\Rightarrow$
(iii) Assume that
$$\begin{eqnarray}\displaystyle \{f\in \unicode[STIX]{x1D6E4}_{0}(X),\text{dom}\,f^{\ast }\subset D\}\subset {\mathcal{E}}^{\unicode[STIX]{x1D711}}. & & \displaystyle\end{eqnarray}$$
Let
$\unicode[STIX]{x1D709}^{\ast }\in D$
. Observe that
$\langle \unicode[STIX]{x1D709}^{\ast },\cdot \rangle \in \unicode[STIX]{x1D6E4}_{0}(X)$
and that
$$\begin{eqnarray}\text{dom}(\langle \unicode[STIX]{x1D709}^{\ast },\cdot \rangle )^{\ast }=\text{dom}\,\unicode[STIX]{x1D6FF}_{\{\unicode[STIX]{x1D709}^{\ast }\}}=\{\unicode[STIX]{x1D709}^{\ast }\}\subset D,\end{eqnarray}$$
hence
$\langle \unicode[STIX]{x1D709}^{\ast },\cdot \rangle \in {\mathcal{E}}^{\unicode[STIX]{x1D711}}$
in view of (12). We then deduce from Proposition 3.3(ii) that
$\unicode[STIX]{x1D709}^{\ast }\in -\text{dom}(-\unicode[STIX]{x1D711})^{\ast }$
. Since this is true for every
$\unicode[STIX]{x1D709}^{\ast }\in D$
, we conclude that
$D\subset -\text{dom}(-\unicode[STIX]{x1D711})^{\ast }$
.
(iii)
$\Rightarrow$
(i) Now assume that
$D\subset -\text{dom}(-\unicode[STIX]{x1D711})^{\ast }$
and let
$f\in \unicode[STIX]{x1D6F4}_{D}^{\ast }$
. There exists
$g:X^{\ast }\rightarrow \overline{\mathbb{R}}$
such that
$f=g^{\ast }$
and
$\text{dom}\,g\subset D$
. The definition of the Legendre–Fenchel conjugate yields
$$\begin{eqnarray}\displaystyle f & = & \displaystyle \sup _{\unicode[STIX]{x1D709}^{\ast }\in X^{\ast }}\{\langle \unicode[STIX]{x1D709}^{\ast },\cdot \rangle -g(\unicode[STIX]{x1D709}^{\ast })\}\nonumber\\ \displaystyle & = & \displaystyle \sup _{\unicode[STIX]{x1D709}^{\ast }\in \text{dom}\,g}\{\langle \unicode[STIX]{x1D709}^{\ast },\cdot \rangle -g(\unicode[STIX]{x1D709}^{\ast })\}.\end{eqnarray}$$
Recalling that
$\text{dom}\,g\subset D\subset -\text{dom}(-\unicode[STIX]{x1D711})^{\ast }$
, we deduce from Proposition 3.3(ii) that the linear function
$\langle \unicode[STIX]{x1D709}^{\ast },\cdot \rangle$
is a
$\unicode[STIX]{x1D711}$
-envelope for every
$\unicode[STIX]{x1D709}^{\ast }\in \text{dom}\,g$
. In view of Proposition 3.2(i), the affine function
$\langle \unicode[STIX]{x1D709}^{\ast },\cdot \rangle -g(\unicode[STIX]{x1D709}^{\ast })$
is also a
$\unicode[STIX]{x1D711}$
-envelope for every
$\unicode[STIX]{x1D709}^{\ast }\in \text{dom}\,g$
. Coming back to formula (13), we infer from Proposition 3.2(ii) that
$f$
is a
$\unicode[STIX]{x1D711}$
-envelope as a supremum of
$\unicode[STIX]{x1D711}$
-envelopes. Finally, we have shown that
$f\in {\mathcal{E}}^{\unicode[STIX]{x1D711}}$
, which proves the inclusion
$\unicode[STIX]{x1D6F4}_{D}^{\ast }\subset {\mathcal{E}}^{\unicode[STIX]{x1D711}}.$
◻
Given a set
$D\subset X^{\ast }$
, the following result explores the links between the class
$\unicode[STIX]{x1D6F4}_{D}^{\ast }$
and the class of functions
$f\in \unicode[STIX]{x1D6E4}_{0}(X)$
satisfying
$\text{dom}\,f^{\ast }\subset D$
. When the set
$D$
is
$w(X^{\ast },X)$
-closed and convex, these classes can be characterized via the support function of
$D$
.
Proposition 3.4. Let
$X$
be a locally convex space and let
$D$
be a non-empty subset of
$X^{\ast }$
.
-
(i) We have
(14)
$$\begin{eqnarray}\displaystyle & \displaystyle \{f\in \unicode[STIX]{x1D6E4}_{0}(X),\text{dom}\,f^{\ast }\subset D\}\cup \{\unicode[STIX]{x1D714}_{X},-\unicode[STIX]{x1D714}_{X}\}\subset \unicode[STIX]{x1D6F4}_{D}^{\ast }, & \displaystyle\end{eqnarray}$$
(15)As a consequence, if the set
$$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6F4}_{D}^{\ast }\subset \{f\in \unicode[STIX]{x1D6E4}_{0}(X),\text{dom}\,f^{\ast }\subset \overline{\text{co}}^{w\ast }(D)\}\cup \{\unicode[STIX]{x1D714}_{X},-\unicode[STIX]{x1D714}_{X}\}. & \displaystyle\end{eqnarray}$$
$D\subset X^{\ast }$
is
$w(X^{\ast },X)$
-closed and convex, the following equality holds true (16)
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6F4}_{D}^{\ast }=\{f\in \unicode[STIX]{x1D6E4}_{0}(X),\text{dom}\,f^{\ast }\subset D\}\cup \{\unicode[STIX]{x1D714}_{X},-\unicode[STIX]{x1D714}_{X}\}. & & \displaystyle\end{eqnarray}$$
-
(ii) If the set
$D\subset X^{\ast }$
is
$w(X^{\ast },X)$
-closed and convex, then (17)
$$\begin{eqnarray}\displaystyle & & \displaystyle \{f\in \unicode[STIX]{x1D6E4}_{0}(X),\text{dom}\,f^{\ast }\subset D\}\nonumber\\ \displaystyle & & \displaystyle \quad =\{f\in \unicode[STIX]{x1D6E4}_{0}(X),f^{\infty }\leqslant \unicode[STIX]{x1D70E}_{D}\}\end{eqnarray}$$
(18)
$$\begin{eqnarray}\displaystyle & & \displaystyle \quad =\{f\in \unicode[STIX]{x1D6E4}_{0}(X),f(y)\leqslant f(x)+\unicode[STIX]{x1D70E}_{D}(y-x),\forall x,y\in X\}.\end{eqnarray}$$
Proof. (i) We have already shown the inclusion
$\{f\in \unicode[STIX]{x1D6E4}_{0}(X),\text{dom}\,f^{\ast }\subset D\}\subset \unicode[STIX]{x1D6F4}_{D}^{\ast }$
, see the proof of Theorem 3.1. On the other hand, we always have
$-\unicode[STIX]{x1D714}_{X}\in \unicode[STIX]{x1D6F4}_{D}^{\ast }$
. Since
$D\neq \emptyset$
, we also have
$\unicode[STIX]{x1D714}_{X}\in \unicode[STIX]{x1D6F4}_{D}^{\ast }$
. This proves the inclusion (14). Let us now establish (15). Assume that
$f\in \unicode[STIX]{x1D6F4}_{D}^{\ast }$
. There exists
$g:X^{\ast }\rightarrow \overline{\mathbb{R}}$
such that
$\text{dom}\,g\subset D$
and
$f=g^{\ast }$
. We distinguish the cases
$\overline{\text{co}}^{w\ast }(g)$
proper and
$\overline{\text{co}}^{w\ast }(g)$
improper. If
$\overline{\text{co}}^{w\ast }(g)=\unicode[STIX]{x1D714}_{X^{\ast }}$
, we have
$g=\unicode[STIX]{x1D714}_{X^{\ast }}$
, hence
$f=-\unicode[STIX]{x1D714}_{X}$
. If
$\overline{\text{co}}^{w\ast }(g)$
takes the value
$-\infty$
, we infer that
$g^{\ast }=(\overline{\text{co}}^{w\ast }(g))^{\ast }=\unicode[STIX]{x1D714}_{X}$
, whence
$f=\unicode[STIX]{x1D714}_{X}$
. Let us now assume that
$\overline{\text{co}}^{w\ast }(g)\in \unicode[STIX]{x1D6E4}_{0}(X^{\ast })$
. It ensues that
$f=g^{\ast }=(\overline{\text{co}}^{w\ast }(g))^{\ast }\in \unicode[STIX]{x1D6E4}_{0}(X)$
. This implies in turn that
$f^{\ast }=\overline{\text{co}}^{w\ast }(g)$
, thus
$$\begin{eqnarray}\text{dom}\,f^{\ast }=\text{dom}(\overline{\text{co}}^{w\ast }(g))\subset \overline{\text{co}}^{w\ast }(\text{dom}\,g)\subset \overline{\text{co}}^{w\ast }(D),\end{eqnarray}$$
which ends the proof of (15). When the set
$D$
is
$w(X^{\ast },X)$
-closed and convex, equality (16) is an immediate consequence of the inclusions (14)–(15).
(ii) Assuming that the set
$D$
is
$w(X^{\ast },X)$
-closed and convex, we have
$\text{dom}\,f^{\ast }\subset D$
if and only if
$\unicode[STIX]{x1D70E}_{\text{dom}\,f^{\ast }}\leqslant \unicode[STIX]{x1D70E}_{D}$
. Recalling that
$\unicode[STIX]{x1D70E}_{\text{dom}\,f^{\ast }}=f^{\infty }$
(see §2), we derive equality (17). Since
$f^{\infty }=\sup _{x\in \text{dom}\,f}(f(\cdot +x)-f(x))$
, we deduce in turn equality (18).◻
If
$X$
is a Banach space and if the set
$D\subset X^{\ast }$
is closed, the class of functions
$f\in \unicode[STIX]{x1D6E4}_{0}(X)$
satisfying
$\text{dom}\,f^{\ast }\subset D$
can be expressed via the subdifferential of
$f$
.
Proposition 3.5. Let
$X$
be a Banach space and let
$D$
be a closed subset of
$X^{\ast }$
. Then we have
$$\begin{eqnarray}\{f\in \unicode[STIX]{x1D6E4}_{0}(X),\text{dom}\,f^{\ast }\subset D\}=\{f\in \unicode[STIX]{x1D6E4}_{0}(X),\unicode[STIX]{x2202}f(x)\subset D\text{ for all }x\in X\}.\end{eqnarray}$$
Proof. Let us first state as a lemma the following direct consequence of the Brønsted–Rockafellar theorem (see [Reference Brønsted and Rockafellar5, Theorem 2]) concerning the conjugate of a function in
$\unicode[STIX]{x1D6E4}_{0}(X)$
.
Lemma 3.1 (See [Reference Brønsted and Rockafellar5, Theorem 2]).
If
$X$
is a Banach space and if
$f\in \unicode[STIX]{x1D6E4}_{0}(X)$
, then
$\text{cl}(\text{dom}\,f^{\ast })=\text{cl}(\text{Rge}(\unicode[STIX]{x2202}f)).$
Assume that the set
$D\subset X^{\ast }$
is closed. From Lemma 3.1, we have for every
$f\in \unicode[STIX]{x1D6E4}_{0}(X)$
,
$$\begin{eqnarray}\displaystyle \text{dom}\,f^{\ast }\subset D & \Longleftrightarrow & \displaystyle \text{Rge}(\unicode[STIX]{x2202}f)\subset D\nonumber\\ \displaystyle & \Longleftrightarrow & \displaystyle \unicode[STIX]{x2202}f(x)\subset D\quad \text{for all }x\in X.\nonumber\end{eqnarray}$$
The announced equality follows immediately. ◻
Applying Theorem 3.1 with particular sets
$D$
, we obtain the following corollaries.
Corollary 3.1. Let
$X$
be a locally convex space and let
$\unicode[STIX]{x1D711}:X\rightarrow \overline{\mathbb{R}}$
be such that
$\unicode[STIX]{x1D711}\neq -\unicode[STIX]{x1D714}_{X}$
. Then the following equivalence holds:
$$\begin{eqnarray}\unicode[STIX]{x1D6E4}(X)\subset {\mathcal{E}}^{\unicode[STIX]{x1D711}}\;\Longleftrightarrow \;\text{dom}(-\unicode[STIX]{x1D711})^{\ast }=X^{\ast }.\end{eqnarray}$$
Proof. It suffices to take
$D=X^{\ast }$
in the equivalence (i)
$\Leftrightarrow$
(iii) of Theorem 3.1.◻
Remark 3.2. Under the assumption
$\text{dom}(-\unicode[STIX]{x1D711})^{\ast }=X^{\ast }$
, the function
$\unicode[STIX]{x1D711}$
cannot be convex (see hereafter). Therefore, the set
${\mathcal{E}}^{\unicode[STIX]{x1D711}}$
is strictly larger than
$\unicode[STIX]{x1D6E4}(X)$
, since it contains the non-convex function
$\unicode[STIX]{x1D711}$
.
If
$\text{dom}(-\unicode[STIX]{x1D711})^{\ast }=X^{\ast }$
, we have
$(-\unicode[STIX]{x1D711})^{\ast }(0)<+\infty$
. Recalling that
$(-\unicode[STIX]{x1D711})^{\ast }(0)=\sup \unicode[STIX]{x1D711}$
, we deduce that the function
$\unicode[STIX]{x1D711}$
is bounded from above on the whole space
$X$
. If, moreover, the function
$\unicode[STIX]{x1D711}$
is convex, we infer from a classical result that it is constant, say
$\unicode[STIX]{x1D711}\equiv \unicode[STIX]{x1D6FD}$
for some
$\unicode[STIX]{x1D6FD}\in \mathbb{R}$
. It ensues that
$(-\unicode[STIX]{x1D711})^{\ast }=\unicode[STIX]{x1D6FD}+\unicode[STIX]{x1D6FF}_{\{0\}}$
, hence
$\text{dom}(-\unicode[STIX]{x1D711})^{\ast }=\{0\}$
, a contradiction. This confirms that functions
$\unicode[STIX]{x1D711}$
with
$\text{dom}(-\unicode[STIX]{x1D711})^{\ast }=X^{\ast }$
cannot be convex.
Given a set
$K\subset X$
, recall that a function
$f:X\rightarrow \mathbb{R}\cup \{+\infty \}$
is said to be
$K$
-non-increasing (respectively
$K$
-non-decreasing) if
$f(y)\leqslant f(x)$
(respectively
$f(y)\geqslant f(x)$
) for all
$x,y\in X$
such that
$y-x\in K$
.
Corollary 3.2. Let
$X$
be a locally convex space. Let
$K\subset X$
be a closed convex cone and let
$\unicode[STIX]{x1D711}:X\rightarrow \overline{\mathbb{R}}$
be such that
$\unicode[STIX]{x1D711}\neq -\unicode[STIX]{x1D714}_{X}$
. Then the set
${\mathcal{E}}^{\unicode[STIX]{x1D711}}$
contains all the functions of
$\unicode[STIX]{x1D6E4}_{0}(X)$
which are
$K$
-non-increasing if and only if
$-K^{\circ }\subset \text{dom}(-\unicode[STIX]{x1D711})^{\ast }$
.
Proof. Take
$D=K^{\circ }$
in the equivalence (ii)
$\Leftrightarrow$
(iii) of Theorem 3.1 to obtain that
$$\begin{eqnarray}\displaystyle \{f\in \unicode[STIX]{x1D6E4}_{0}(X),\text{dom}\,f^{\ast }\subset K^{\circ }\}\subset {\mathcal{E}}^{\unicode[STIX]{x1D711}} & \Longleftrightarrow & \displaystyle K^{\circ }\subset -\text{dom}(-\unicode[STIX]{x1D711})^{\ast }\nonumber\\ \displaystyle & \Longleftrightarrow & \displaystyle -K^{\circ }\subset \text{dom}(-\unicode[STIX]{x1D711})^{\ast }.\end{eqnarray}$$
On the other hand, observe by (18) that for
$f\in \unicode[STIX]{x1D6E4}_{0}(X)$
,
$$\begin{eqnarray}\displaystyle \text{dom}\,f^{\ast }\subset K^{\circ } & \Longleftrightarrow & \displaystyle f(y)\leqslant f(x)+\unicode[STIX]{x1D70E}_{K^{\circ }}(y-x)\quad \text{for all }x,y\in X\nonumber\\ \displaystyle & \Longleftrightarrow & \displaystyle f(y)\leqslant f(x)+\unicode[STIX]{x1D6FF}_{K}(y-x)\quad \text{for all }x,y\in X\nonumber\\ \displaystyle & \Longleftrightarrow & \displaystyle f\text{ is }K\text{-non-increasing.}\end{eqnarray}$$
The announced equivalence then follows immediately from (19) and (20). ◻
In the following, when
$X$
is a normed space we will denote by
$\mathbb{B}_{X}$
(respectively
$\mathbb{B}_{X^{\ast }}$
) the closed unit ball of
$X$
(respectively
$X^{\ast }$
).
Corollary 3.3. Let
$(X,\Vert \cdot \Vert )$
be a normed space. Let a real
$k\geqslant 0$
and let
$\unicode[STIX]{x1D711}:X\rightarrow \overline{\mathbb{R}}$
be such that
$\unicode[STIX]{x1D711}\neq -\unicode[STIX]{x1D714}_{X}$
. Then the set
${\mathcal{E}}^{\unicode[STIX]{x1D711}}$
contains all the functions of
$\unicode[STIX]{x1D6E4}_{0}(X)$
which are
$k$
-Lipschitz continuous on
$X$
if and only if
$k\mathbb{B}_{X^{\ast }}\subset \text{dom}(-\unicode[STIX]{x1D711})^{\ast }$
.
Proof. Take
$D=k\mathbb{B}_{X^{\ast }}$
in the equivalence (ii)
$\Leftrightarrow$
(iii) of Theorem 3.1 to obtain that
$$\begin{eqnarray}\displaystyle \{f\in \unicode[STIX]{x1D6E4}_{0}(X),\text{dom}\,f^{\ast }\subset k\mathbb{B}_{X^{\ast }}\}\subset {\mathcal{E}}^{\unicode[STIX]{x1D711}} & \Longleftrightarrow & \displaystyle k\mathbb{B}_{X^{\ast }}\subset -\text{dom}(-\unicode[STIX]{x1D711})^{\ast }\nonumber\\ \displaystyle & \Longleftrightarrow & \displaystyle k\mathbb{B}_{X^{\ast }}\subset \text{dom}(-\unicode[STIX]{x1D711})^{\ast }.\end{eqnarray}$$
Then observe by (18) that for
$f\in \unicode[STIX]{x1D6E4}_{0}(X)$
,
$$\begin{eqnarray}\displaystyle \text{dom}\,f^{\ast }\subset k\mathbb{B}_{X^{\ast }} & \Longleftrightarrow & \displaystyle f(y)\leqslant f(x)+k\Vert y-x\Vert \quad \text{for all }x,y\in X\nonumber\\ \displaystyle & \Longleftrightarrow & \displaystyle f\text{ is }k\text{-Lipschitz on }X,\end{eqnarray}$$
where the last equivalence is obtained by reversing the roles of
$x$
and
$y$
. The announced equivalence then follows immediately from (21) and (22).◻
4 Equivalence between functions and sets
Recall that for
$f:X\rightarrow \overline{\mathbb{R}}$
, the epigraph (respectively hypograph) of
$f$
is defined by
$$\begin{eqnarray}\displaystyle & & \displaystyle \text{epi}\,f=\{(x,\unicode[STIX]{x1D706})\in X\times \mathbb{R},f(x)\leqslant \unicode[STIX]{x1D706}\}\nonumber\\ \displaystyle & & \displaystyle \quad (\text{respectively }\text{hypo}\,f=\{(x,\unicode[STIX]{x1D706})\in X\times \mathbb{R},f(x)\geqslant \unicode[STIX]{x1D706}\}).\nonumber\end{eqnarray}$$
The following result allows us to express the epigraph of
$f^{\unicode[STIX]{x1D711}}$
as an intersection of sets which are translated from the epigraph of
$\unicode[STIX]{x1D711}$
.
Proposition 4.1. Let
$X$
be a vector space and let
$\unicode[STIX]{x1D711}:X\rightarrow \overline{\mathbb{R}}$
.
-
(i) For every
$f:X\rightarrow \overline{\mathbb{R}}$
, we have
$$\begin{eqnarray}\text{epi}\,f^{\unicode[STIX]{x1D711}}=\mathop{\bigcap }_{u\in \text{hypo}(-f)}u+\text{epi}\,\unicode[STIX]{x1D711}.\end{eqnarray}$$
-
(ii) For every
$g:X\rightarrow \overline{\mathbb{R}}$
, the following equivalence holds:
$$\begin{eqnarray}g\in {\mathcal{E}}^{\unicode[STIX]{x1D711}}\;\Longleftrightarrow \;\text{epi}\,g=\mathop{\bigcap }_{u\in U}u+\text{epi}\,\unicode[STIX]{x1D711}\quad \text{ for some }U\subset X\times \mathbb{R}.\end{eqnarray}$$
Proof. (i) For every
$x\in X$
and
$\unicode[STIX]{x1D706}\in \mathbb{R}$
, the following equivalences hold true:
The announced equality follows.
(ii) Let
$g:X\rightarrow \overline{\mathbb{R}}$
. If
$g\in {\mathcal{E}}^{\unicode[STIX]{x1D711}}$
, there exists
$f:X\rightarrow \overline{\mathbb{R}}$
such that
$g=f^{\unicode[STIX]{x1D711}}$
. In view of (i), we obtain that
$\text{epi}\,g=\bigcap _{u\in U}u+\text{epi}\,\unicode[STIX]{x1D711}$
with
$U=\text{hypo}(-f)$
. Conversely, assume that
$\text{epi}\,g=\bigcap _{u\in U}u+\text{epi}\,\unicode[STIX]{x1D711}$
for some
$U\subset X\times \mathbb{R}$
. Then we have
$$\begin{eqnarray}\displaystyle \text{epi}\,g & = & \displaystyle \mathop{\bigcap }_{(x,\unicode[STIX]{x1D706})\in U}(x,\unicode[STIX]{x1D706})+\text{epi}\,\unicode[STIX]{x1D711}\nonumber\\ \displaystyle & = & \displaystyle \mathop{\bigcap }_{(x,\unicode[STIX]{x1D706})\in U}\text{epi}[\unicode[STIX]{x1D711}(\cdot -x)+\unicode[STIX]{x1D706}]\nonumber\\ \displaystyle & = & \displaystyle \text{epi}\Bigl[\sup _{(x,\unicode[STIX]{x1D706})\in U}\unicode[STIX]{x1D711}(\cdot -x)+\unicode[STIX]{x1D706}\Bigr].\nonumber\end{eqnarray}$$
Hence, we deduce that
$g=\sup _{(x,\unicode[STIX]{x1D706})\in U}(\unicode[STIX]{x1D711}(\cdot -x)+\unicode[STIX]{x1D706})$
, which shows by (i) and (ii) in Proposition 3.2 that
$g\in {\mathcal{E}}^{\unicode[STIX]{x1D711}}$
.◻
Given a set
$\unicode[STIX]{x1D6EC}\subset X$
, the previous result suggests to consider the class
${\mathcal{I}}^{\unicode[STIX]{x1D6EC}}$
of subsets of
$X$
defined as followsFootnote
1
$$\begin{eqnarray}{\mathcal{I}}^{\unicode[STIX]{x1D6EC}}=\{C^{\unicode[STIX]{x1D6EC}},C\subset X\},\quad \text{where }C^{\unicode[STIX]{x1D6EC}}=\mathop{\bigcap }_{x\in C}x+\unicode[STIX]{x1D6EC}.\end{eqnarray}$$
By conventionFootnote
2
, we take
$\emptyset ^{\unicode[STIX]{x1D6EC}}=\bigcap _{x\in \emptyset }x+\unicode[STIX]{x1D6EC}=X$
for every set
$\unicode[STIX]{x1D6EC}\subset X$
. This implies that
$X\in {\mathcal{I}}^{\unicode[STIX]{x1D6EC}}$
for every
$\unicode[STIX]{x1D6EC}\subset X$
. It is immediate to check that
${\mathcal{I}}^{X}=\{X\}$
, while
${\mathcal{I}}^{\emptyset }=\{\emptyset ,X\}$
. A set
$D\subset X$
belongs to the class
${\mathcal{I}}^{\unicode[STIX]{x1D6EC}}$
if it is equal to some intersection of translated sets from
$\unicode[STIX]{x1D6EC}$
. It ensues immediately that the class
${\mathcal{I}}^{\unicode[STIX]{x1D6EC}}$
is stable under translation and intersection.
Example 4.1. Take
$r>0$
and
$\unicode[STIX]{x1D6EC}=r\mathbb{B}_{X}$
. The class
${\mathcal{I}}^{r\mathbb{B}_{X}}$
corresponds to the class studied by Vial [Reference Vial36] under the terminology of
$r$
-strongly convex sets. More generally, for a closed convex set
$\unicode[STIX]{x1D6EC}\subset X$
, the sets of the form
$C^{\unicode[STIX]{x1D6EC}}$
are called
$\unicode[STIX]{x1D6EC}$
-strongly convex. The
$\unicode[STIX]{x1D6EC}$
-strongly convex sets are thoroughly studied by Polovinkin [Reference Polovinkin25], under an additional condition on the set
$\unicode[STIX]{x1D6EC}$
(which is assumed to be generating, see [Reference Polovinkin25] for more details).
The definition of
$C^{\unicode[STIX]{x1D6EC}}$
is directly linked to the star-difference of sets. For every
$C_{1}$
,
$C_{2}\subset X$
, the star-difference of
$C_{1}$
with
$C_{2}$
is the set 
given by
We deduce immediately from the above definition that 
for every
$C$
,
$\unicode[STIX]{x1D6EC}\subset X$
. The star-difference of sets was used in [Reference Pshenichnyi27] in the context of differential games. See also [Reference Hiriart-Urruty12] for the links between the star-difference of sets and the deconvolution operation, also called epigraphical star-difference.
Given
$C\subset X$
and
$\unicode[STIX]{x1D6EC}\subset X$
, the next proposition gives several expressions for the set
$C^{\unicode[STIX]{x1D6EC}}$
.
Proposition 4.2. Let
$X$
be a vector space. For any sets
$C\subset X$
and
$\unicode[STIX]{x1D6EC}\subset X$
, we have:
-
(i)
$C^{\unicode[STIX]{x1D6EC}}=\{x\in X,x-C\subset \unicode[STIX]{x1D6EC}\}=\{x\in X,C\subset x-\unicode[STIX]{x1D6EC}\}$
; -
(ii)
$X\setminus C^{\unicode[STIX]{x1D6EC}}=C+(X\setminus \unicode[STIX]{x1D6EC})$
or equivalently
$C^{X\setminus \unicode[STIX]{x1D6EC}}=X\setminus (C+\unicode[STIX]{x1D6EC})$
; -
(iii)
$(X\setminus \unicode[STIX]{x1D6EC})^{X\setminus C}=C^{\unicode[STIX]{x1D6EC}}$
.
Proof. (i) It suffices to observe that
$$\begin{eqnarray}\displaystyle x\in C^{\unicode[STIX]{x1D6EC}} & \Longleftrightarrow & \displaystyle \text{for all }u\in C,x\in u+\unicode[STIX]{x1D6EC}\nonumber\\ \displaystyle & \Longleftrightarrow & \displaystyle \text{for all }u\in C,x-u\in \unicode[STIX]{x1D6EC}\nonumber\\ \displaystyle & \Longleftrightarrow & \displaystyle x-C\subset \unicode[STIX]{x1D6EC}\nonumber\\ \displaystyle & \Longleftrightarrow & \displaystyle C\subset x-\unicode[STIX]{x1D6EC}.\nonumber\end{eqnarray}$$
(ii) From the definition of
$C^{\unicode[STIX]{x1D6EC}}$
, we deduce immediately that
$$\begin{eqnarray}X\setminus C^{\unicode[STIX]{x1D6EC}}=\mathop{\bigcup }_{u\in C}u+(X\setminus \unicode[STIX]{x1D6EC})=C+(X\setminus \unicode[STIX]{x1D6EC}),\end{eqnarray}$$
which is the first equality in (ii). From this equality with
$X\setminus \unicode[STIX]{x1D6EC}$
in place of
$\unicode[STIX]{x1D6EC}$
, we obtain that
$X\setminus C^{X\setminus \unicode[STIX]{x1D6EC}}=C+\unicode[STIX]{x1D6EC}$
, or equivalently
$C^{X\setminus \unicode[STIX]{x1D6EC}}=X\setminus (C+\unicode[STIX]{x1D6EC})$
.
(iii) We infer from the previous assertion that
$$\begin{eqnarray}X\setminus [(X\setminus \unicode[STIX]{x1D6EC})^{X\setminus C}]=(X\setminus \unicode[STIX]{x1D6EC})+C=X\setminus C^{\unicode[STIX]{x1D6EC}},\end{eqnarray}$$
whence the equality
$(X\setminus \unicode[STIX]{x1D6EC})^{X\setminus C}=C^{\unicode[STIX]{x1D6EC}}$
.◻
The elements
$D$
of
${\mathcal{I}}^{\unicode[STIX]{x1D6EC}}$
can be characterized by the equality
$(D^{-\unicode[STIX]{x1D6EC}})^{\unicode[STIX]{x1D6EC}}=D$
. This is the subject of the next proposition.
Proposition 4.3. Let
$X$
be a vector space and let
$\unicode[STIX]{x1D6EC}\subset X$
. For any set
$D\subset X$
, the set
$(D^{-\unicode[STIX]{x1D6EC}})^{\unicode[STIX]{x1D6EC}}$
is the smallest element of
${\mathcal{I}}^{\unicode[STIX]{x1D6EC}}$
containing the set
$D$
. As a consequence, the following equivalence holds true:
$$\begin{eqnarray}D\in {\mathcal{I}}^{\unicode[STIX]{x1D6EC}}\;\Longleftrightarrow \;(D^{-\unicode[STIX]{x1D6EC}})^{\unicode[STIX]{x1D6EC}}=D.\end{eqnarray}$$
Proof. Let
$S$
be the subset of
$X$
defined by
$$\begin{eqnarray}S=\mathop{\bigcap }_{x\in X,x+\unicode[STIX]{x1D6EC}\supset D}x+\unicode[STIX]{x1D6EC}.\end{eqnarray}$$
We clearly have
$S\in {\mathcal{I}}^{\unicode[STIX]{x1D6EC}}$
and
$S\supset D$
. Now let any
$S^{\prime }\in {\mathcal{I}}^{\unicode[STIX]{x1D6EC}}$
with
$S^{\prime }\supset D$
. By definition, there exists some
$C\subset X$
such that
$S^{\prime }=\bigcap _{x\in C}x+\unicode[STIX]{x1D6EC}$
. The inclusion
$S^{\prime }\supset D$
implies that
$x+\unicode[STIX]{x1D6EC}\supset D$
for every
$x\in C$
and therefore
$$\begin{eqnarray}S^{\prime }=\mathop{\bigcap }_{x\in C}x+\unicode[STIX]{x1D6EC}\supset \mathop{\bigcap }_{x\in X,x+\unicode[STIX]{x1D6EC}\supset D}x+\unicode[STIX]{x1D6EC}=S.\end{eqnarray}$$
This proves that the set
$S$
is the smallest element of
${\mathcal{I}}^{\unicode[STIX]{x1D6EC}}$
containing
$D$
. Recall now from Proposition 4.2(i) that condition
$x+\unicode[STIX]{x1D6EC}\supset D$
is equivalent to
$x\in D^{-\unicode[STIX]{x1D6EC}}$
. We deduce that
$$\begin{eqnarray}S=\mathop{\bigcap }_{x\in D^{-\unicode[STIX]{x1D6EC}}}x+\unicode[STIX]{x1D6EC}=(D^{-\unicode[STIX]{x1D6EC}})^{\unicode[STIX]{x1D6EC}}.\end{eqnarray}$$
This finishes the proof of the first assertion. The second assertion is an immediate consequence of the first. ◻
Let us write the expression of the double envelope
$(D^{-\unicode[STIX]{x1D6EC}})^{\unicode[STIX]{x1D6EC}}$
by using the star-difference operation
In view of Proposition 4.2, the complement of the set
$(D^{-\unicode[STIX]{x1D6EC}})^{\unicode[STIX]{x1D6EC}}$
can be expressed as
From equalities (23)–(24) and Proposition 4.3, we deduce that
The last equality amounts to saying that the set
$X\setminus D$
is exactly
$(X\setminus \unicode[STIX]{x1D6EC})$
-regular in the sense of [Reference Martinez-Legaz and Penot18].
With the notation introduced above, for
$f$
,
$g:X\rightarrow \overline{\mathbb{R}}$
, the results of Proposition 4.1 can be restated as
$$\begin{eqnarray}\text{epi}\,f^{\unicode[STIX]{x1D711}}=(\text{hypo}(-f))^{\text{epi}\,\unicode[STIX]{x1D711}}\end{eqnarray}$$
and
$$\begin{eqnarray}g\in {\mathcal{E}}^{\unicode[STIX]{x1D711}}\;\Longleftrightarrow \;\text{epi}\,g\in {\mathcal{I}}^{\text{epi}\,\unicode[STIX]{x1D711}}.\end{eqnarray}$$
This shows that the study of
$\unicode[STIX]{x1D711}$
-envelopes amounts to that of the class
${\mathcal{I}}^{\text{epi}\,\unicode[STIX]{x1D711}}$
. Conversely, given a set
$\unicode[STIX]{x1D6EC}\subset X$
, the class
${\mathcal{I}}^{\unicode[STIX]{x1D6EC}}$
can be fully described via the
$\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D6EC}}$
-envelopes.
Proposition 4.4. Let
$X$
be a vector space and let
$\unicode[STIX]{x1D6EC}\subset X$
.
-
(i) For every function
$f:X\rightarrow \overline{\mathbb{R}}$
, we haveFootnote
1
(25)As a consequence, the equality
$$\begin{eqnarray}\displaystyle f^{\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D6EC}}}=-\!\inf _{X}f+\unicode[STIX]{x1D6FF}_{(\text{dom}\,f)^{\unicode[STIX]{x1D6EC}}}. & & \displaystyle\end{eqnarray}$$
$(\unicode[STIX]{x1D6FF}_{C})^{\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D6EC}}}=\unicode[STIX]{x1D6FF}_{C^{\unicode[STIX]{x1D6EC}}}$
holds for any non-empty set
$C\subset X$
.
-
(ii) For every function
$g:X\rightarrow \overline{\mathbb{R}}$
such that
$g\neq \pm \unicode[STIX]{x1D714}_{X}$
, we have
$$\begin{eqnarray}g\in {\mathcal{E}}^{\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D6EC}}}\;\Longleftrightarrow \;g=\unicode[STIX]{x1D6FD}+\unicode[STIX]{x1D6FF}_{C^{\unicode[STIX]{x1D6EC}}}\quad \text{for some }\unicode[STIX]{x1D6FD}\in \mathbb{R}\text{ and some }C\neq \emptyset .\end{eqnarray}$$
Proof. (i) For every function
$f:X\rightarrow \overline{\mathbb{R}}$
and every
$x\in X$
, the definition of
$f^{\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D6EC}}}$
gives
First assume that
$x-\text{dom}\,f\subset \unicode[STIX]{x1D6EC}$
. For every
$y\in \text{dom}\,f$
, we then have
$x-y\in \unicode[STIX]{x1D6EC}$
, whence
$\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D6EC}}(x-y)=0$
. It ensues that
$$\begin{eqnarray}f^{\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D6EC}}}(x)=\sup _{y\in \text{dom}\,f}-f(y)=\sup _{X}(-f)=-\!\inf _{X}f.\end{eqnarray}$$
Now assume that
$x-\text{dom}\,f\not \subset \unicode[STIX]{x1D6EC}$
. In this case, there exists
$y\in \text{dom}\,f$
such that
$x-y\notin \unicode[STIX]{x1D6EC}$
. We then have
$\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D6EC}}(x-y)=+\infty$
, whence
$f^{\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D6EC}}}(x)=+\infty$
. Finally, we obtain for every
$x\in X$
,
$$\begin{eqnarray}f^{\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D6EC}}}(x)=\left\{\begin{array}{@{}ll@{}}\displaystyle -\!\inf _{X}f & \text{if }x-\text{dom}\,f\subset \unicode[STIX]{x1D6EC},\\ \displaystyle +\infty & \text{otherwise.}\end{array}\right.\end{eqnarray}$$
Condition
$x-\text{dom}\,f\subset \unicode[STIX]{x1D6EC}$
is equivalent to
$x\in (\text{dom}\,f)^{\unicode[STIX]{x1D6EC}}$
in view of Proposition 4.2(i). Formula (25) follows immediately. For the last assertion, it suffices to take
$f=\unicode[STIX]{x1D6FF}_{C}$
.
(ii) Let
$g\in {\mathcal{E}}^{\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D6EC}}}$
be such that
$g\neq \pm \unicode[STIX]{x1D714}_{X}$
. There exists
$f:X\rightarrow \overline{\mathbb{R}}$
such that
$g=f^{\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D6EC}}}$
, hence we deduce from (i) that
$g=-\!\inf _{X}f+\unicode[STIX]{x1D6FF}_{(\text{dom}\,f)^{\unicode[STIX]{x1D6EC}}}$
. Since
$g\neq \pm \unicode[STIX]{x1D714}_{X}$
, we have
$\inf _{X}f\in \mathbb{R}$
and
$\text{dom}\,f\neq \emptyset$
. It suffices then to take
$\unicode[STIX]{x1D6FD}=-\inf _{X}f$
and
$C=\text{dom}\,f$
. Conversely, assume that
$g=\unicode[STIX]{x1D6FD}+\unicode[STIX]{x1D6FF}_{C^{\unicode[STIX]{x1D6EC}}}$
for some
$\unicode[STIX]{x1D6FD}\in \mathbb{R}$
and some
$C\neq \emptyset$
. Assertion (i) then shows that
$g=f^{\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D6EC}}}$
for the function
$f$
defined by
$f=-\unicode[STIX]{x1D6FD}+\unicode[STIX]{x1D6FF}_{C}$
, hence
$g\in {\mathcal{E}}^{\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D6EC}}}$
.◻
Remark 4.1. The previous proposition shows that for every
$C$
,
$\unicode[STIX]{x1D6EC}\subset X$
with
$C\neq \emptyset$
,
$$\begin{eqnarray}\displaystyle (\unicode[STIX]{x1D6FF}_{C})^{\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D6EC}}}=(-\unicode[STIX]{x1D6FF}_{C})\triangle \unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D6EC}}=\unicode[STIX]{x1D6FF}_{C^{\unicode[STIX]{x1D6EC}}}. & & \displaystyle\end{eqnarray}$$
It is interesting to compare this formula with the following one
$$\begin{eqnarray}\displaystyle (\unicode[STIX]{x1D6FF}_{C})^{-\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D6EC}}}=(-\unicode[STIX]{x1D6FF}_{C})\triangle (-\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D6EC}})=-\unicode[STIX]{x1D6FF}_{C+\unicode[STIX]{x1D6EC}}, & & \displaystyle\end{eqnarray}$$
that is obtained as a consequence of the equality
$\unicode[STIX]{x1D6FF}_{C+\unicode[STIX]{x1D6EC}}=\unicode[STIX]{x1D6FF}_{C}{\bigtriangledown}\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D6EC}}$
.
Corollary 4.1. Let
$X$
be a vector space. For every set
$\unicode[STIX]{x1D6EC}\subset X$
and every set
$D\subset X$
such that
$D\neq \emptyset$
and
$D\neq X$
, the following equivalence holds:
$$\begin{eqnarray}\unicode[STIX]{x1D6FF}_{D}\in {\mathcal{E}}^{\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D6EC}}}\;\Longleftrightarrow \;D\in {\mathcal{I}}^{\unicode[STIX]{x1D6EC}}.\end{eqnarray}$$
In fact, the implication from the left to the right is true as soon as
$D\neq \emptyset$
, while the reverse one is true if
$D\neq X$
.
Proof. First assume that
$\unicode[STIX]{x1D6FF}_{D}\in {\mathcal{E}}^{\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D6EC}}}$
and that
$D\neq \emptyset$
. There exists
$f:X\rightarrow \overline{\mathbb{R}}$
such that
$\unicode[STIX]{x1D6FF}_{D}=f^{\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D6EC}}}$
, hence we deduce from Proposition 4.4(i) that
$\unicode[STIX]{x1D6FF}_{D}=-\!\inf _{X}f+\unicode[STIX]{x1D6FF}_{(\text{dom}\,f)^{\unicode[STIX]{x1D6EC}}}$
. Since
$D\neq \emptyset$
, we have
$\inf _{X}f=0$
and
$D=(\text{dom}\,f)^{\unicode[STIX]{x1D6EC}}\in {\mathcal{I}}^{\unicode[STIX]{x1D6EC}}$
. Conversely, assume that
$D\in {\mathcal{I}}^{\unicode[STIX]{x1D6EC}}$
and that
$D\neq X$
. This implies that
$D=C^{\unicode[STIX]{x1D6EC}}$
for some
$C\neq \emptyset$
, and hence by Proposition 4.4(i) again
$\unicode[STIX]{x1D6FF}_{D}=\unicode[STIX]{x1D6FF}_{C^{\unicode[STIX]{x1D6EC}}}=(\unicode[STIX]{x1D6FF}_{C})^{\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D6EC}}}\in {\mathcal{E}}^{\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D6EC}}}.$
◻
Let us now study the class
${\mathcal{E}}^{-\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D6EC}}}$
. From the generalized conjugation point of view, the case
$\unicode[STIX]{x1D711}=-\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D6EC}}$
is a special instance of a coupling functional
$c:X\times Y\rightarrow \overline{\mathbb{R}}$
of the type
$c=-\unicode[STIX]{x1D6FF}_{G}$
, where
$G$
is a subset of
$X\times Y$
. The corresponding conjugation operator, which arises in quasiconvex analysis, has been considered in many papers, see, for example, [Reference Martinez-Legaz and Cambini17, Reference Singer33, Reference Volle37].
Proposition 4.5. Let
$X$
be a vector space. Let
$\unicode[STIX]{x1D6EC}$
be a non-empty subset of
$X$
and let
$f:X\rightarrow \overline{\mathbb{R}}$
. Then:
-
(i) we have
(28)this means equivalently that for every
$$\begin{eqnarray}\displaystyle f\in {\mathcal{E}}^{-\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D6EC}}}\;\Longleftrightarrow \;f=\sup _{y\in \unicode[STIX]{x1D6EC}}\inf _{z\in \unicode[STIX]{x1D6EC}}f(\cdot -y+z); & & \displaystyle\end{eqnarray}$$
$x\in X$
and every
$\unicode[STIX]{x1D706}<f(x)$
, there exists
$y\in \unicode[STIX]{x1D6EC}$
such that
$f(x-y+z)\geqslant \unicode[STIX]{x1D706}$
for every
$z\in \unicode[STIX]{x1D6EC}$
;
-
(ii) if
$f\in {\mathcal{E}}^{-\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D6EC}}}$
and if
$\unicode[STIX]{x1D6EC}+\unicode[STIX]{x1D6EC}\subset \unicode[STIX]{x1D6EC}$
, then
$f$
is
$\unicode[STIX]{x1D6EC}$
-non-decreasing; conversely, if
$f$
is
$\unicode[STIX]{x1D6EC}$
-non-decreasing and if
$0\in \unicode[STIX]{x1D6EC}$
, then
$f\in {\mathcal{E}}^{-\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D6EC}}}$
.
Proof. (i) The equivalence (7)
$\Longleftrightarrow$
(8) yields
$$\begin{eqnarray}f\in {\mathcal{E}}^{-\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D6EC}}}\;\Longleftrightarrow \;f=(f^{(-\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D6EC}})_{-}})^{-\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D6EC}}}.\end{eqnarray}$$
On the other hand, we have
and, hence,
$$\begin{eqnarray}(f^{(-\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D6EC}})_{-}})^{-\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D6EC}}}=\sup _{y\in \unicode[STIX]{x1D6EC}}-f^{(-\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D6EC}})_{-}}(\cdot -y)=\sup _{y\in \unicode[STIX]{x1D6EC}}\inf _{z\in \unicode[STIX]{x1D6EC}}f(\cdot -y+z).\end{eqnarray}$$
We deduce immediately the equivalence (28).
Since the inequality
$(f^{(-\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D6EC}})_{-}})^{-\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D6EC}}}\leqslant f$
is always satisfied, we infer that
$f\in {\mathcal{E}}^{-\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D6EC}}}$
if and only if for every
$x\in X$
,
$$\begin{eqnarray}\sup _{y\in \unicode[STIX]{x1D6EC}}\inf _{z\in \unicode[STIX]{x1D6EC}}f(x-y+z)\geqslant f(x).\end{eqnarray}$$
The last assertion of (i) follows immediately.
(ii) Assume that
$f\in {\mathcal{E}}^{-\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D6EC}}}$
and that
$\unicode[STIX]{x1D6EC}+\unicode[STIX]{x1D6EC}\subset \unicode[STIX]{x1D6EC}$
. Let
$\unicode[STIX]{x1D709}\in \unicode[STIX]{x1D6EC}$
. In view of (28), we have for every
$x\in X$
,
$$\begin{eqnarray}\displaystyle f(x+\unicode[STIX]{x1D709}) & = & \displaystyle \sup _{y\in \unicode[STIX]{x1D6EC}}\inf _{z\in \unicode[STIX]{x1D6EC}}f(x+\unicode[STIX]{x1D709}-y+z)\nonumber\\ \displaystyle & = & \displaystyle \sup _{y\in \unicode[STIX]{x1D6EC}}\inf _{z^{\prime }\in \unicode[STIX]{x1D709}+\unicode[STIX]{x1D6EC}}f(x-y+z^{\prime })\nonumber\\ \displaystyle & {\geqslant} & \displaystyle \sup _{y\in \unicode[STIX]{x1D6EC}}\inf _{z^{\prime }\in \unicode[STIX]{x1D6EC}}f(x-y+z^{\prime })\quad \text{since }\unicode[STIX]{x1D709}+\unicode[STIX]{x1D6EC}\subset \unicode[STIX]{x1D6EC}\nonumber\\ \displaystyle & = & \displaystyle f(x).\nonumber\end{eqnarray}$$
Since this is true for every
$\unicode[STIX]{x1D709}\in \unicode[STIX]{x1D6EC}$
, we infer that
$f$
is
$\unicode[STIX]{x1D6EC}$
-non-decreasing.
Conversely, assume that
$f$
is
$\unicode[STIX]{x1D6EC}$
-non-decreasing and that
$0\in \unicode[STIX]{x1D6EC}$
. For every
$y,z\in \unicode[STIX]{x1D6EC}$
, we have
$$\begin{eqnarray}f(\cdot -y)\leqslant f(\cdot -y+z)\leqslant f(\cdot +z).\end{eqnarray}$$
It ensues immediately that
$$\begin{eqnarray}\sup _{y\in \unicode[STIX]{x1D6EC}}f(\cdot -y)\leqslant \sup _{y\in \unicode[STIX]{x1D6EC}}\inf _{z\in \unicode[STIX]{x1D6EC}}f(\cdot -y+z)\leqslant \inf _{z\in \unicode[STIX]{x1D6EC}}f(\cdot +z).\end{eqnarray}$$
Since
$0\in \unicode[STIX]{x1D6EC}$
, we obtain
$\sup _{y\in \unicode[STIX]{x1D6EC}}f(\cdot -y)=\inf _{z\in \unicode[STIX]{x1D6EC}}f(\cdot +z)=f$
, and hence
$f=\sup _{y\in \unicode[STIX]{x1D6EC}}\inf _{z\in \unicode[STIX]{x1D6EC}}f(\cdot -y+z)$
. In view of (28), we conclude that
$f\in {\mathcal{E}}^{-\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D6EC}}}$
.◻
Remark 4.2. When
$\unicode[STIX]{x1D6EC}+\unicode[STIX]{x1D6EC}\subset \unicode[STIX]{x1D6EC}$
and
$0\in \unicode[STIX]{x1D6EC}$
, the equivalence
$$\begin{eqnarray}f\in {\mathcal{E}}^{-\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D6EC}}}\;\Longleftrightarrow \;f\text{ is }\unicode[STIX]{x1D6EC}\text{-non-decreasing}\end{eqnarray}$$
can be recovered by using the subadditivity of the function
$\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D6EC}}$
, see §6.
For a function
$f:X\rightarrow \overline{\mathbb{R}}$
and
$r\in \overline{\mathbb{R}}$
, the notation
$[f\geqslant r]$
(respectively
$[f>r]$
) denotes the set
$\{x\in X,f(x)\geqslant r\}$
(respectively
$\{x\in X,f(x)>r\}$
). We adopt the corresponding notation for the sublevel sets.
Proposition 4.6. Given a vector space
$X$
and
$\unicode[STIX]{x1D6EC}\subset X$
, let
$f$
,
$g:X\rightarrow \overline{\mathbb{R}}$
be extended real-valued functions.
-
(i) For every
$\unicode[STIX]{x1D706}\in \overline{\mathbb{R}}$
, we have
$$\begin{eqnarray}[f^{-\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D6EC}}}\leqslant \unicode[STIX]{x1D706}]=[f<-\unicode[STIX]{x1D706}]^{X\setminus \unicode[STIX]{x1D6EC}}.\end{eqnarray}$$
-
(ii) Assume that for every
$\unicode[STIX]{x1D706}\in \overline{\mathbb{R}}$
, there exists
$C_{\unicode[STIX]{x1D706}}\subset X$
such that Then we have
$$\begin{eqnarray}[g\leqslant \unicode[STIX]{x1D706}]=C_{\unicode[STIX]{x1D706}}^{X\setminus \unicode[STIX]{x1D6EC}}.\end{eqnarray}$$
$g=(-h)^{-\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D6EC}}}$
for the function
$h:X\rightarrow \overline{\mathbb{R}}$
defined by
$h(x)=\sup \{\unicode[STIX]{x1D706}\in \overline{\mathbb{R}},x\in C_{\unicode[STIX]{x1D706}}\}$
.
-
(iii) The following assertions are equivalent:
-
(a)
$g\in {\mathcal{E}}^{-\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D6EC}}}$
; -
(b) for all
$\unicode[STIX]{x1D706}\in \overline{\mathbb{R}}$
,
$[g\leqslant \unicode[STIX]{x1D706}]\in {\mathcal{I}}^{X\setminus \unicode[STIX]{x1D6EC}}$
.
-
Proof. (i) For every
$\unicode[STIX]{x1D706}\in \overline{\mathbb{R}}$
and
$x\in X$
, the following equivalences hold:
We deduce the equality
$[f^{-\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D6EC}}}\leqslant \unicode[STIX]{x1D706}]=[f<-\unicode[STIX]{x1D706}]^{X\setminus \unicode[STIX]{x1D6EC}}.$
(ii) Fix
$x\in X$
and observe that
$$\begin{eqnarray}\displaystyle g(x)=\sup \{\unicode[STIX]{x1D706}\in \overline{\mathbb{R}},g(x)>\unicode[STIX]{x1D706}\}. & & \displaystyle\end{eqnarray}$$
By assumption, we have for every
$\unicode[STIX]{x1D706}\in \overline{\mathbb{R}}$
,
$$\begin{eqnarray}[g\leqslant \unicode[STIX]{x1D706}]=\mathop{\bigcap }_{y\in C_{\unicode[STIX]{x1D706}}}y+X\setminus \unicode[STIX]{x1D6EC},\end{eqnarray}$$
hence
$[g>\unicode[STIX]{x1D706}]=\bigcup _{y\in C_{\unicode[STIX]{x1D706}}}y+\unicode[STIX]{x1D6EC}$
. From (29), we infer that
Since this is true for every
$x\in X$
, we conclude that
$g=(-h)^{-\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D6EC}}}$
.
(iii) The implication (a)
$\Longrightarrow$
(b) follows immediately from (i), while the converse implication is a consequence of (ii).◻
Corollary 4.2. Let
$X$
be a vector space and let
$\unicode[STIX]{x1D6EC}\subset X$
. Assume that the function
$g:X\rightarrow \overline{\mathbb{R}}$
is such that
$g(X)=\{\unicode[STIX]{x1D6FC}_{1},\ldots ,\unicode[STIX]{x1D6FC}_{n}\}\subset \overline{\mathbb{R}}$
. If
$\inf g=-\infty$
, the assertions (a) and (b) of Proposition 4.6(iii) are equivalent to the following:
-
$(\text{b}^{\prime })$
-
for all
$i\in \{1,\ldots ,n\}$
,
$[g\leqslant \unicode[STIX]{x1D6FC}_{i}]\in {\mathcal{I}}^{X\setminus \unicode[STIX]{x1D6EC}}$
;
while if
$\inf g>-\infty$
, they are equivalent to:
-
$(\text{b}^{\prime \prime })$
-
for all
$i\in \{1,\ldots ,n\}$
,
$[g\leqslant \unicode[STIX]{x1D6FC}_{i}]\in {\mathcal{I}}^{X\setminus \unicode[STIX]{x1D6EC}}\text{and }\unicode[STIX]{x1D6EC}\neq \emptyset$
.
Proof. If
$\inf g=-\infty$
, assertion (b) of Proposition 4.6(iii) is clearly equivalent to (b
$^{\prime }$
), while if
$\inf g>-\infty$
, one has to add the complementary condition
$\emptyset \in {\mathcal{I}}^{X\setminus \unicode[STIX]{x1D6EC}}$
. Then observe that
$$\begin{eqnarray}\emptyset \in {\mathcal{I}}^{X\setminus \unicode[STIX]{x1D6EC}}\Longleftrightarrow X^{X\setminus \unicode[STIX]{x1D6EC}}=\emptyset \Longleftrightarrow X+\unicode[STIX]{x1D6EC}=X\Longleftrightarrow \unicode[STIX]{x1D6EC}\neq \emptyset ,\end{eqnarray}$$
and hence the equivalence (b)
$\Longleftrightarrow$
(b
$^{\prime \prime }$
) follows.◻
Corollary 4.3. Given a vector space
$X$
, let
$\unicode[STIX]{x1D6EC}\subset X$
and
$D\subset X$
. The following equivalences hold:
-
(i)
$-\unicode[STIX]{x1D6FF}_{D}\in {\mathcal{E}}^{-\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D6EC}}}\Longleftrightarrow X\setminus D\in {\mathcal{I}}^{X\setminus \unicode[STIX]{x1D6EC}}$
; -
(ii)
$\unicode[STIX]{x1D6FF}_{D}\in {\mathcal{E}}^{-\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D6EC}}}\Longleftrightarrow D\in {\mathcal{I}}^{X\setminus \unicode[STIX]{x1D6EC}}$
and
$\unicode[STIX]{x1D6EC}\neq \emptyset$
.
Proof. For (i) (respectively (ii)) it suffices to apply Corollary 4.2 with the function
$g=-\unicode[STIX]{x1D6FF}_{D}$
(respectively
$g=\unicode[STIX]{x1D6FF}_{D}$
), which takes the values
$\unicode[STIX]{x1D6FC}_{1}=-\infty$
,
$\unicode[STIX]{x1D6FC}_{2}=0$
(respectively
$\unicode[STIX]{x1D6FC}_{1}=0$
,
$\unicode[STIX]{x1D6FC}_{2}=+\infty$
).◻
Combining Corollaries 4.1 and 4.3, we derive the following result giving various characterizations of
${\mathcal{I}}^{\unicode[STIX]{x1D6EC}}$
via the classes
${\mathcal{E}}^{\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D6EC}}}$
and
${\mathcal{E}}^{-\unicode[STIX]{x1D6FF}_{X\setminus \unicode[STIX]{x1D6EC}}}$
.
Corollary 4.4. For every set
$\unicode[STIX]{x1D6EC}\subset X$
and every set
$D\subset X$
such that
$D\neq \emptyset$
and
$D\neq X$
, the following equivalences hold:
$$\begin{eqnarray}D\in {\mathcal{I}}^{\unicode[STIX]{x1D6EC}}\;\Longleftrightarrow \;\unicode[STIX]{x1D6FF}_{D}\in {\mathcal{E}}^{\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D6EC}}}\;\Longleftrightarrow \;-\unicode[STIX]{x1D6FF}_{X\setminus D}\in {\mathcal{E}}^{-\unicode[STIX]{x1D6FF}_{X\setminus \unicode[STIX]{x1D6EC}}}\;\Longleftrightarrow \;\unicode[STIX]{x1D6FF}_{D}\in {\mathcal{E}}^{-\unicode[STIX]{x1D6FF}_{X\setminus \unicode[STIX]{x1D6EC}}}.\end{eqnarray}$$
Proof. The first equivalence is a consequence of Corollary 4.1, under the assumptions
$D\neq \emptyset$
and
$D\neq X$
. The equivalence
$D\in {\mathcal{I}}^{\unicode[STIX]{x1D6EC}}\;\Longleftrightarrow \;-\unicode[STIX]{x1D6FF}_{X\setminus D}\in {\mathcal{E}}^{-\unicode[STIX]{x1D6FF}_{X\setminus \unicode[STIX]{x1D6EC}}}$
follows from Corollary 4.3(i) applied with
$X\setminus D$
(respectively
$X\setminus \unicode[STIX]{x1D6EC}$
) in place of
$D$
(respectively
$\unicode[STIX]{x1D6EC}$
). If
$\unicode[STIX]{x1D6EC}\neq X$
, the equivalence
$D\in {\mathcal{I}}^{\unicode[STIX]{x1D6EC}}\;\Longleftrightarrow \;\unicode[STIX]{x1D6FF}_{D}\in {\mathcal{E}}^{-\unicode[STIX]{x1D6FF}_{X\setminus \unicode[STIX]{x1D6EC}}}$
is a consequence of Corollary 4.3(ii) applied with
$X\setminus \unicode[STIX]{x1D6EC}$
in place of
$\unicode[STIX]{x1D6EC}$
. If
$\unicode[STIX]{x1D6EC}=X$
, the equivalence becomes
$D\in {\mathcal{I}}^{X}\;\Longleftrightarrow \;\unicode[STIX]{x1D6FF}_{D}\in {\mathcal{E}}^{-\unicode[STIX]{x1D714}_{X}}$
. Since
${\mathcal{I}}^{X}=\{X\}$
and
${\mathcal{E}}^{-\unicode[STIX]{x1D714}_{X}}=\{-\unicode[STIX]{x1D714}_{X}\}$
, the equivalence amounts to
$D=X\;\Longleftrightarrow \;\unicode[STIX]{x1D6FF}_{D}=-\unicode[STIX]{x1D714}_{X}$
. The condition
$D=X$
is not realized by assumption, while the condition
$\unicode[STIX]{x1D6FF}_{D}=-\unicode[STIX]{x1D714}_{X}$
is never realized. It ensues that the equivalence trivially holds true if
$\unicode[STIX]{x1D6EC}=X$
.◻
By combining Proposition 3.3(ii) and Proposition 4.6(iii), we obtain the following statement.
Proposition 4.7. Let
$X$
be a locally convex space. Let
$\unicode[STIX]{x1D6EC}\subset X$
and
$\unicode[STIX]{x1D709}^{\ast }\in X^{\ast }$
. If
$\unicode[STIX]{x1D6EC}\neq X$
, the following assertions are equivalent:
-
(a)
$[\langle \unicode[STIX]{x1D709}^{\ast },\cdot \rangle \leqslant 0]\in {\mathcal{I}}^{\unicode[STIX]{x1D6EC}}$
; -
(b) for all
$\unicode[STIX]{x1D706}\in \mathbb{R},[\langle \unicode[STIX]{x1D709}^{\ast },\cdot \rangle \leqslant \unicode[STIX]{x1D706}]\in {\mathcal{I}}^{\unicode[STIX]{x1D6EC}}$
; -
(c)
$\langle \unicode[STIX]{x1D709}^{\ast },\cdot \rangle \in {\mathcal{E}}^{-\unicode[STIX]{x1D6FF}_{X\setminus \unicode[STIX]{x1D6EC}}}$
; -
(d)
$\unicode[STIX]{x1D709}^{\ast }\in -\text{dom}\,\unicode[STIX]{x1D70E}_{X\setminus \unicode[STIX]{x1D6EC}}$
.
Proof. For every
$\unicode[STIX]{x1D706}\in \mathbb{R}$
, the set
$[\langle \unicode[STIX]{x1D709}^{\ast },\cdot \rangle \leqslant \unicode[STIX]{x1D706}]$
is translated from the set
$[\langle \unicode[STIX]{x1D709}^{\ast },\cdot \rangle \leqslant 0]$
. Since the class
${\mathcal{I}}^{\unicode[STIX]{x1D6EC}}$
is stable under translations, the equivalence (a)
$\Longleftrightarrow$
(b) follows immediately. Recalling that
$\unicode[STIX]{x1D6EC}\neq X$
, we have
$\emptyset \in {\mathcal{I}}^{\unicode[STIX]{x1D6EC}}$
, hence the inclusion in (b) holds for
$\unicode[STIX]{x1D706}=-\infty$
. Since
$X\in {\mathcal{I}}^{\unicode[STIX]{x1D6EC}}$
, the inclusion in (b) is also satisfied for
$\unicode[STIX]{x1D706}=+\infty$
. The equivalence (b)
$\Longleftrightarrow$
(c) can then be deduced from Proposition 4.6(iii) applied with
$X\setminus \unicode[STIX]{x1D6EC}$
in place of
$\unicode[STIX]{x1D6EC}$
. Finally, the equivalence (c)
$\Longleftrightarrow$
(d) follows from Proposition 3.3(ii) applied with
$\unicode[STIX]{x1D711}=-\unicode[STIX]{x1D6FF}_{X\setminus \unicode[STIX]{x1D6EC}}$
.◻
Let us denote by
${\mathcal{C}}(X)$
the class of non-empty closed convex sets of
$X$
.
Theorem 4.1. Let
$X$
be a locally convex space. Let
$\unicode[STIX]{x1D6EC}\subset X$
be such that
$\unicode[STIX]{x1D6EC}\neq X$
. For every cone
$Q\subset X^{\ast }$
, the following equivalence holds true:
$$\begin{eqnarray}\{C\in {\mathcal{C}}(X),\text{dom}\,\unicode[STIX]{x1D70E}_{C}\subset Q\}\subset {\mathcal{I}}^{\unicode[STIX]{x1D6EC}}\;\Longleftrightarrow \;Q\subset -\text{dom}\,\unicode[STIX]{x1D70E}_{X\setminus \unicode[STIX]{x1D6EC}}.\end{eqnarray}$$
Proof. Let
$Q\subset X^{\ast }$
be a cone and assume that
$$\begin{eqnarray}\displaystyle \{C\in {\mathcal{C}}(X),\text{dom}\,\unicode[STIX]{x1D70E}_{C}\subset Q\}\subset {\mathcal{I}}^{\unicode[STIX]{x1D6EC}}. & & \displaystyle\end{eqnarray}$$
Let
$\unicode[STIX]{x1D709}^{\ast }\in Q$
. Setting
$C=[\langle \unicode[STIX]{x1D709}^{\ast },\cdot \rangle \leqslant 0]\in {\mathcal{C}}(X)$
, we have
$\unicode[STIX]{x1D70E}_{C}=\unicode[STIX]{x1D6FF}_{\mathbb{R}_{+}\unicode[STIX]{x1D709}^{\ast }}$
, and hence
$\text{dom}\,\unicode[STIX]{x1D70E}_{C}=\mathbb{R}_{+}\unicode[STIX]{x1D709}^{\ast }\subset Q$
. In view of (30), it ensues that
$C\in {\mathcal{I}}^{\unicode[STIX]{x1D6EC}}$
. We then deduce from Proposition 4.7 that
$\unicode[STIX]{x1D709}^{\ast }\in -\text{dom}\,\unicode[STIX]{x1D70E}_{X\setminus \unicode[STIX]{x1D6EC}}$
. Since this is true for every
$\unicode[STIX]{x1D709}^{\ast }\in Q$
, we conclude that
$Q\subset -\text{dom}\,\unicode[STIX]{x1D70E}_{X\setminus \unicode[STIX]{x1D6EC}}$
.
Now assume that
$Q\subset -\text{dom}\,\unicode[STIX]{x1D70E}_{X\setminus \unicode[STIX]{x1D6EC}}$
and let
$C\in {\mathcal{C}}(X)$
be such that
$\text{dom}\,\unicode[STIX]{x1D70E}_{C}\subset Q$
. Then
$\unicode[STIX]{x1D6FF}_{C}\in \unicode[STIX]{x1D6E4}_{0}(X)$
with
$\text{dom}\,\unicode[STIX]{x1D6FF}_{C}^{\ast }\subset Q$
, and since
$$\begin{eqnarray}Q\subset -\text{dom}\,\unicode[STIX]{x1D70E}_{X\setminus \unicode[STIX]{x1D6EC}}=-\text{dom}\,\unicode[STIX]{x1D6FF}_{X\setminus \unicode[STIX]{x1D6EC}}^{\ast }=-\text{dom}(-(-\unicode[STIX]{x1D6FF}_{X\setminus \unicode[STIX]{x1D6EC}}))^{\ast },\end{eqnarray}$$
by Theorem 3.1 we have
$\unicode[STIX]{x1D6FF}_{C}\in {\mathcal{E}}^{-\unicode[STIX]{x1D6FF}_{X\setminus \unicode[STIX]{x1D6EC}}}$
(keep in mind
$-\unicode[STIX]{x1D6FF}_{X\setminus \unicode[STIX]{x1D6EC}}\neq -\unicode[STIX]{x1D714}_{X}$
since
$\unicode[STIX]{x1D6EC}\neq X$
). Corollary 4.3(ii) yields that
$C\in {\mathcal{I}}^{\unicode[STIX]{x1D6EC}}$
as desired. Finally, we have shown the inclusion (30), which ends the proof.◻
Applying Theorem 4.1 with
$Q=X^{\ast }$
, we immediately obtain the following result.
Corollary 4.5. Let
$X$
be a locally convex space. Let
$\unicode[STIX]{x1D6EC}\subset X$
be such that
$\unicode[STIX]{x1D6EC}\neq X$
. Then, the following equivalence holds true:
$$\begin{eqnarray}{\mathcal{C}}(X)\subset {\mathcal{I}}^{\unicode[STIX]{x1D6EC}}\;\Longleftrightarrow \;\text{dom}\,\unicode[STIX]{x1D70E}_{X\setminus \unicode[STIX]{x1D6EC}}=X^{\ast }.\end{eqnarray}$$
5 A preorder relation on
${\mathcal{F}}(X,\overline{\mathbb{R}})$
based on
$\unicode[STIX]{x1D711}$
-envelopes
Let
$X$
be a vector space and let
${\mathcal{F}}(X,\overline{\mathbb{R}})$
be the set of extended real-valued functions on
$X$
. We define the relation
${\sim}$
on the space
${\mathcal{F}}(X,\overline{\mathbb{R}})$
as follows: for every
$\unicode[STIX]{x1D711}$
,
$\unicode[STIX]{x1D713}:X\rightarrow \overline{\mathbb{R}}$
,
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D713}\sim \unicode[STIX]{x1D711} & \Longleftrightarrow & \displaystyle \text{there exist }\unicode[STIX]{x1D709}\in X\text{ and }\unicode[STIX]{x1D6FC}\in \mathbb{R}\text{ such that }\unicode[STIX]{x1D713}=\unicode[STIX]{x1D711}(\cdot -\unicode[STIX]{x1D709})+\unicode[STIX]{x1D6FC}\nonumber\\ \displaystyle & \Longleftrightarrow & \displaystyle \unicode[STIX]{x1D713}\text{is a }\unicode[STIX]{x1D711}\text{-elementary function}.\nonumber\end{eqnarray}$$
Clearly, the relation
${\sim}$
is reflexive, symmetric and transitive, hence defines an equivalence relation. The objective of this section is to determine suitableFootnote
1
subsets
${\mathcal{G}}$
of
${\mathcal{F}}(X,\overline{\mathbb{R}})$
such that the following implication holds true for every
$\unicode[STIX]{x1D711}$
,
$\unicode[STIX]{x1D713}\in {\mathcal{G}}$
:
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D713}\in {\mathcal{E}}^{\unicode[STIX]{x1D711}}\quad \text{and}\quad \unicode[STIX]{x1D711}\in {\mathcal{E}}^{\unicode[STIX]{x1D713}}\;\Longrightarrow \;\unicode[STIX]{x1D713}\sim \unicode[STIX]{x1D711}. & & \displaystyle\end{eqnarray}$$
5.1 The coercive case
For any function
$\unicode[STIX]{x1D711}:X\rightarrow \overline{\mathbb{R}}$
, the deconvolution function
$\unicode[STIX]{x1D711}\ominus \unicode[STIX]{x1D711}$
defined by 
can be expressed as a
$\unicode[STIX]{x1D711}$
-envelope via the equality
$\unicode[STIX]{x1D711}\ominus \unicode[STIX]{x1D711}=(\unicode[STIX]{x1D711}_{-})^{\unicode[STIX]{x1D711}}$
. The next lemma shows that this function is subadditive. Recall that a function
$f:X\rightarrow \overline{\mathbb{R}}$
is said to be subadditive if for any
$x$
,
$y\in X$
,
Lemma 5.1. Let
$X$
be a vector space and let
$f$
,
$\unicode[STIX]{x1D711}:X\rightarrow \overline{\mathbb{R}}$
. For any
$x$
,
$x^{\prime }\in X$
, we have
Moreover, the function
$\unicode[STIX]{x1D711}\ominus \unicode[STIX]{x1D711}$
is subadditive.
Proof. Fix
$x,x^{\prime }\in X$
. It is immediate to check that for every
$y\in X$
,
Taking the supremum over
$y\in X$
and using [Reference Moreau21, Proposition 4.a] we deduce that
which yields the desired inequality. Further taking
$f=\unicode[STIX]{x1D711}_{-}$
in the above inequality and using the identity
$(\unicode[STIX]{x1D711}_{-})^{\unicode[STIX]{x1D711}}=\unicode[STIX]{x1D711}\ominus \unicode[STIX]{x1D711}$
, we obtain
hence the function
$\unicode[STIX]{x1D711}\ominus \unicode[STIX]{x1D711}$
is subadditive.◻
If the space
$(X,\Vert \cdot \Vert )$
is normed and if the function
$\unicode[STIX]{x1D711}$
satisfies the coercivity property
$\lim _{\Vert x\Vert \rightarrow +\infty }\unicode[STIX]{x1D711}(x)/\Vert x\Vert =+\infty$
, the following lemma shows that
$\unicode[STIX]{x1D711}\ominus \unicode[STIX]{x1D711}=+\infty$
on
$X\setminus \{0\}$
.
Lemma 5.2. Let
$(X,\Vert \cdot \Vert )$
be a normed space and let
$\unicode[STIX]{x1D711}:X\rightarrow \overline{\mathbb{R}}$
be an extended real-valued function. Assume that
$\unicode[STIX]{x1D711}\neq +\unicode[STIX]{x1D714}_{X}$
and
$\lim _{\Vert x\Vert \rightarrow +\infty }\unicode[STIX]{x1D711}(x)/\Vert x\Vert =+\infty$
. Then we have
$\unicode[STIX]{x1D711}\ominus \unicode[STIX]{x1D711}=+\infty$
on
$X\setminus \{0\}$
.
Proof. Let us argue by contradiction and assume that there exists
$u\neq 0$
such that
$(\unicode[STIX]{x1D711}\ominus \unicode[STIX]{x1D711})(u)<+\infty$
. Let us fix
$\overline{x}\in \text{dom}\,\unicode[STIX]{x1D711}$
and observe that for everyFootnote
1
$n\in \mathbb{N}$
,
It ensues that
$$\begin{eqnarray}\frac{1}{n}\unicode[STIX]{x1D711}(\overline{x}+nu)\leqslant \frac{1}{n}\unicode[STIX]{x1D711}(\overline{x})+(\unicode[STIX]{x1D711}\ominus \unicode[STIX]{x1D711})(u),\end{eqnarray}$$
and taking the upper limit as
$n\rightarrow +\infty$
, we deduce that
$$\begin{eqnarray}\limsup _{n\rightarrow +\infty }\frac{1}{n}\unicode[STIX]{x1D711}(\overline{x}+nu)\leqslant (\unicode[STIX]{x1D711}\ominus \unicode[STIX]{x1D711})(u),\end{eqnarray}$$
which contradicts the fact that
$\lim _{\Vert x\Vert \rightarrow +\infty }\unicode[STIX]{x1D711}(x)/\Vert x\Vert =+\infty$
. Finally, we obtain that
$\unicode[STIX]{x1D711}\ominus \unicode[STIX]{x1D711}=+\infty$
on
$X\setminus \{0\}$
.◻
Theorem 5.1. Let
$X$
be a vector space and let
$\unicode[STIX]{x1D711}$
,
$\unicode[STIX]{x1D713}:X\rightarrow \overline{\mathbb{R}}$
be such that
$\unicode[STIX]{x1D713}\in {\mathcal{E}}^{\unicode[STIX]{x1D711}}$
and
$\unicode[STIX]{x1D711}\in {\mathcal{E}}^{\unicode[STIX]{x1D713}}$
.
-
(i) If
$\unicode[STIX]{x1D711}\ominus \unicode[STIX]{x1D711}=+\infty$
on
$X\setminus \{0\}$
, then we have
$\unicode[STIX]{x1D713}\sim \unicode[STIX]{x1D711}$
. -
(ii) Assume that
$(X,\Vert \cdot \Vert )$
is a normed space. If
$\lim _{\Vert x\Vert \rightarrow +\infty }\unicode[STIX]{x1D711}(x)/\Vert x\Vert =+\infty$
(respectively
$\lim _{\Vert x\Vert \rightarrow +\infty }\unicode[STIX]{x1D711}(x)/\Vert x\Vert =-\infty$
), then we have
$\unicode[STIX]{x1D713}\sim \unicode[STIX]{x1D711}$
.
Proof. If
$\unicode[STIX]{x1D711}=\pm \unicode[STIX]{x1D714}_{X}$
, it is immediate to check that
$\unicode[STIX]{x1D713}=\unicode[STIX]{x1D711}$
. From now on, let us assume that
$\unicode[STIX]{x1D711}\neq \pm \unicode[STIX]{x1D714}_{X}$
. Since
$\unicode[STIX]{x1D713}\in {\mathcal{E}}^{\unicode[STIX]{x1D711}}$
and
$\unicode[STIX]{x1D711}\in {\mathcal{E}}^{\unicode[STIX]{x1D713}}$
, there exist
$f$
,
$g:X\rightarrow \overline{\mathbb{R}}$
such that
$-\unicode[STIX]{x1D713}=(-\unicode[STIX]{x1D711}){\bigtriangledown}f$
and
$-\unicode[STIX]{x1D711}=(-\unicode[STIX]{x1D713}){\bigtriangledown}g$
. It ensues that
$$\begin{eqnarray}\displaystyle -\unicode[STIX]{x1D711}=(-\unicode[STIX]{x1D711}){\bigtriangledown}(f{\bigtriangledown}g). & & \displaystyle\end{eqnarray}$$
Now observe that
(i) Assume that
$\unicode[STIX]{x1D711}\ominus \unicode[STIX]{x1D711}=+\infty$
on
$X\setminus \{0\}$
. We then deduce from the above inequality that
$$\begin{eqnarray}\displaystyle f{\bigtriangledown}g=+\infty \quad \text{on }X\setminus \{0\}. & & \displaystyle\end{eqnarray}$$
If
$f{\bigtriangledown}g=\unicode[STIX]{x1D714}_{X}$
, we infer from (32) that
$\unicode[STIX]{x1D711}=-\unicode[STIX]{x1D714}_{X}$
, thus implying in turn that
$\unicode[STIX]{x1D713}=-\unicode[STIX]{x1D714}_{X}$
. If
$f{\bigtriangledown}g\neq \unicode[STIX]{x1D714}_{X}$
, equality (33) shows that
$\text{dom}(f{\bigtriangledown}g)=\{0\}$
. Recalling that
$\text{dom}(f{\bigtriangledown}g)=\text{dom}\,f+\text{dom}\,g$
, we deduce that
$\text{dom}\,f+\text{dom}\,g=\{0\}$
. Hence, there exists
$\unicode[STIX]{x1D709}\in X$
such that
$\text{dom}\,f=\{\unicode[STIX]{x1D709}\}$
and
$\text{dom}\,g=\{-\unicode[STIX]{x1D709}\}$
. We infer that
and
If
$f(\unicode[STIX]{x1D709})\in \mathbb{R}$
, we obtain from (34) that
$\unicode[STIX]{x1D713}=\unicode[STIX]{x1D711}(\cdot -\unicode[STIX]{x1D709})-f(\unicode[STIX]{x1D709})$
and therefore
$\unicode[STIX]{x1D713}\sim \unicode[STIX]{x1D711}$
. If
$g(-\unicode[STIX]{x1D709})\in \mathbb{R}$
, equality (35) shows that
$\unicode[STIX]{x1D711}=\unicode[STIX]{x1D713}(\cdot +\unicode[STIX]{x1D709})-g(-\unicode[STIX]{x1D709})$
, and hence
$\unicode[STIX]{x1D711}\sim \unicode[STIX]{x1D713}$
. On the other hand, if
$f(\unicode[STIX]{x1D709})=g(-\unicode[STIX]{x1D709})=-\infty$
, we deduce from (34)–(35) that
$$\begin{eqnarray}-\unicode[STIX]{x1D713}\leqslant (-\unicode[STIX]{x1D711})(\cdot -\unicode[STIX]{x1D709})\quad \text{and}\quad -\unicode[STIX]{x1D711}\leqslant (-\unicode[STIX]{x1D713})(\cdot +\unicode[STIX]{x1D709}),\end{eqnarray}$$
thus implying that
$\unicode[STIX]{x1D713}=\unicode[STIX]{x1D711}(\cdot -\unicode[STIX]{x1D709})$
and therefore
$\unicode[STIX]{x1D713}\sim \unicode[STIX]{x1D711}$
.
(ii) First assume that
$\lim _{\Vert x\Vert \rightarrow +\infty }\unicode[STIX]{x1D711}(x)/\Vert x\Vert =+\infty$
. We infer from Lemma 5.2 that
$\unicode[STIX]{x1D711}\ominus \unicode[STIX]{x1D711}=+\infty$
on
$X\setminus \{0\}$
and we conclude with (i).
Now assume that
$\lim _{\Vert x\Vert \rightarrow +\infty }\unicode[STIX]{x1D711}(x)/\Vert x\Vert =-\infty$
. From Lemma 5.2, we deduce that
$(-\unicode[STIX]{x1D711})\ominus (-\unicode[STIX]{x1D711})=+\infty$
on
$X\setminus \{0\}$
. Recalling that
$$\begin{eqnarray}(-\unicode[STIX]{x1D711})\ominus (-\unicode[STIX]{x1D711})=(-\unicode[STIX]{x1D711}_{-})^{-\unicode[STIX]{x1D711}}=\unicode[STIX]{x1D711}^{\unicode[STIX]{x1D711}_{-}}=[(\unicode[STIX]{x1D711}_{-})^{\unicode[STIX]{x1D711}}]_{-}=[\unicode[STIX]{x1D711}\ominus \unicode[STIX]{x1D711}]_{-},\end{eqnarray}$$
we infer that
$\unicode[STIX]{x1D711}\ominus \unicode[STIX]{x1D711}=+\infty$
on
$X\setminus \{0\}$
and we conclude again with (i).◻
Let us define the relation
$\preccurlyeq$
on
${\mathcal{F}}(X,\overline{\mathbb{R}})$
by
$$\begin{eqnarray}\unicode[STIX]{x1D713}\preccurlyeq \unicode[STIX]{x1D711}\;\Longleftrightarrow \;\unicode[STIX]{x1D713}\in {\mathcal{E}}^{\unicode[STIX]{x1D711}}.\end{eqnarray}$$
The relation
$\preccurlyeq$
is clearly reflexive, and also transitive in view of Proposition 3.2(iii). It is compatible with the equivalence relation
${\sim}$
, i.e.
$$\begin{eqnarray}\unicode[STIX]{x1D711}\sim \unicode[STIX]{x1D711}^{\prime },\quad \unicode[STIX]{x1D713}\sim \unicode[STIX]{x1D713}^{\prime }\quad \text{and}\quad \unicode[STIX]{x1D713}\preccurlyeq \unicode[STIX]{x1D711}\;\Longrightarrow \;\unicode[STIX]{x1D713}^{\prime }\preccurlyeq \unicode[STIX]{x1D711}^{\prime }.\end{eqnarray}$$
It ensues that we can properly define the relation
$\preccurlyeq$
on the quotient set
${\mathcal{F}}(X,\overline{\mathbb{R}})/\sim$
. The relation
$\preccurlyeq$
so defined on
${\mathcal{F}}(X,\overline{\mathbb{R}})/\sim$
is clearly reflexive and transitive, hence it is a preorder. Let us denote by
${\mathcal{G}}$
,
${\mathcal{G}}^{\prime }$
and
${\mathcal{G}}^{\prime \prime }$
the following respective sets:
$$\begin{eqnarray}\displaystyle {\mathcal{G}} & = & \displaystyle \{f:X\rightarrow \overline{\mathbb{R}},f\ominus f=+\infty \text{on }X\setminus \{0\}\},\nonumber\\ \displaystyle {\mathcal{G}}^{\prime } & = & \displaystyle \Bigl\{f:X\rightarrow \overline{\mathbb{R}},\lim _{\Vert x\Vert \rightarrow +\infty }f(x)/\Vert x\Vert =+\infty \Bigr\},\nonumber\\ \displaystyle {\mathcal{G}}^{\prime \prime } & = & \displaystyle \Bigl\{f:X\rightarrow \overline{\mathbb{R}},\lim _{\Vert x\Vert \rightarrow +\infty }f(x)/\Vert x\Vert =-\infty \Bigr\}.\nonumber\end{eqnarray}$$
Theorem 5.1 expresses that for every
$\unicode[STIX]{x1D711}$
,
$\unicode[STIX]{x1D713}\in {\mathcal{G}}$
(respectively
${\mathcal{G}}^{\prime }$
,
${\mathcal{G}}^{\prime \prime }$
), we have
$$\begin{eqnarray}\unicode[STIX]{x1D713}\preccurlyeq \unicode[STIX]{x1D711},\qquad \unicode[STIX]{x1D711}\preccurlyeq \unicode[STIX]{x1D713}\quad \Longrightarrow \quad \unicode[STIX]{x1D713}\sim \unicode[STIX]{x1D711}.\end{eqnarray}$$
Hence, the induced relation
$\preccurlyeq$
on the quotient set
${\mathcal{G}}/\sim$
(respectively
${\mathcal{G}}^{\prime }/\sim$
,
${\mathcal{G}}^{\prime \prime }/\sim$
) is antisymmetric, thus giving rise to an order relation.
Let us now specialize the result of Theorem 5.1 in the case of sets. We define the equivalence relation
${\sim}$
on
${\mathcal{P}}(X)$
by
$$\begin{eqnarray}C\sim D\;\Longleftrightarrow \;\text{ there exists }\unicode[STIX]{x1D709}\in X\text{ such that }D=C+\unicode[STIX]{x1D709},\end{eqnarray}$$
along with the preorder relation
$\preccurlyeq$
on
${\mathcal{P}}(X)$
by
$$\begin{eqnarray}C\preccurlyeq D\;\Longleftrightarrow \;C\in {\mathcal{I}}^{D}.\end{eqnarray}$$
Recall that the star-difference 
is defined by
By applying Theorem 5.1 with indicator functions, we obtain the following corollary.
Corollary 5.1. Let
$X$
be a vector space and let
$\unicode[STIX]{x1D6E4}$
,
$\unicode[STIX]{x1D6E5}\subset X$
be such that
$\unicode[STIX]{x1D6E5}\in {\mathcal{I}}^{\unicode[STIX]{x1D6E4}}$
and
$\unicode[STIX]{x1D6E4}\in {\mathcal{I}}^{\unicode[STIX]{x1D6E5}}$
.
-
(i) If
then we have
$\unicode[STIX]{x1D6E5}\sim \unicode[STIX]{x1D6E4}$
. -
(ii) Assume that
$(X,\Vert \cdot \Vert )$
is a normed space. If the set
$\unicode[STIX]{x1D6E4}$
(respectively
$X\setminus \unicode[STIX]{x1D6E4}$
) is bounded, then we have
$\unicode[STIX]{x1D6E5}\sim \unicode[STIX]{x1D6E4}$
.
then we have 
Proof. If
$\unicode[STIX]{x1D6E4}\in \{\emptyset ,X\}$
(respectively
$\unicode[STIX]{x1D6E5}\in \{\emptyset ,X\}$
), it is immediate to check that
$\unicode[STIX]{x1D6E5}=\unicode[STIX]{x1D6E4}$
. Let us now assume that
$\unicode[STIX]{x1D6E4}\notin \{\emptyset ,X\}$
and
$\unicode[STIX]{x1D6E5}\notin \{\emptyset ,X\}$
. In view of Corollary 4.1, the assumptions
$\unicode[STIX]{x1D6E5}\in {\mathcal{I}}^{\unicode[STIX]{x1D6E4}}$
and
$\unicode[STIX]{x1D6E4}\in {\mathcal{I}}^{\unicode[STIX]{x1D6E5}}$
imply that
$\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D6E5}}\in {\mathcal{E}}^{\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D6E4}}}$
and
$\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D6E4}}\in {\mathcal{E}}^{\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D6E5}}}$
.
(i) Assume that 
Then, by (5) and Proposition 4.4(i), we have
By applying Theorem 5.1(i) with
$\unicode[STIX]{x1D711}=\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D6E4}}$
and
$\unicode[STIX]{x1D713}=\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D6E5}}$
, we obtain that
$\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D6E5}}\sim \unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D6E4}}$
and hence
$\unicode[STIX]{x1D6E5}\sim \unicode[STIX]{x1D6E4}$
.
(ii) First assume that
$\unicode[STIX]{x1D6E4}$
is bounded. Then the indicator function
$\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D6E4}}$
is coercive and we deduce from Lemma 5.2 that
$\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D6E4}}\ominus \unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D6E4}}=+\infty$
on
$X\setminus \{0\}$
. This implies that 
and we conclude with (i). Now assume that
$X\setminus \unicode[STIX]{x1D6E4}$
is bounded. From what precedes, we have 
Observing that
we infer that 
and we conclude again with (i). ◻
Let us denote by
${\mathcal{Q}}$
,
${\mathcal{Q}}^{\prime }$
and
${\mathcal{Q}}^{\prime \prime }$
the following respective sets:
The above corollary expresses that for every
$\unicode[STIX]{x1D6E4}$
,
$\unicode[STIX]{x1D6E5}\in {\mathcal{Q}}$
(respectively
${\mathcal{Q}}^{\prime }$
,
${\mathcal{Q}}^{\prime \prime }$
), we have
$$\begin{eqnarray}\unicode[STIX]{x1D6E5}\preccurlyeq \unicode[STIX]{x1D6E4},\quad \unicode[STIX]{x1D6E4}\preccurlyeq \unicode[STIX]{x1D6E5}\;\Longrightarrow \;\unicode[STIX]{x1D6E5}\sim \unicode[STIX]{x1D6E4}.\end{eqnarray}$$
Hence, the induced relation
$\preccurlyeq$
on the quotient set
${\mathcal{Q}}/\sim$
(respectively
${\mathcal{Q}}^{\prime }/\sim$
,
${\mathcal{Q}}^{\prime \prime }/\sim$
) is antisymmetric, thus giving rise to an order relation.
5.2 The case
$\unicode[STIX]{x1D711}$
,
$\unicode[STIX]{x1D713}\in -\unicode[STIX]{x1D6E4}_{0}(X)$
Let us first state a result that will be a key ingredient for the next theorem.
Lemma 5.3. Let
$X$
be a vector space, let
$D\subset X$
be a convex set and let us denote by
$\text{Aff}(D)$
the affine space generated by
$D$
. Assume that a real-valued function
$h:D\rightarrow \mathbb{R}$
is both convex and concave. Then there exists a unique affine function
$\tilde{h}:\text{Aff}(D)\rightarrow \mathbb{R}$
such that
$\tilde{h}_{|D}=h$
.
For a proof of this result, the reader is referred to [Reference Vesely35]. By extending affinely the function
$\tilde{h}$
to the whole space
$X$
, we deduce from the above result that there exists a linear function
$\ell :X\rightarrow \mathbb{R}$
along with
$\unicode[STIX]{x1D6FC}\in \mathbb{R}$
such that
$h=\ell _{|D}+\unicode[STIX]{x1D6FC}$
.
In view of stating the next theorem, given a locally convex space
$X$
recall that the Mackey topology
$\unicode[STIX]{x1D70F}(X^{\ast },X)$
on
$X^{\ast }$
is defined as the finest locally convex topology
${\mathcal{T}}$
on
$X^{\ast }$
such that the topological dual of
$(X^{\ast },{\mathcal{T}})$
coincides with
$X$
. If
$(X,\Vert \cdot \Vert )$
is normed, this topology is exactly that associated with the dual norm
$\Vert \cdot \Vert _{X^{\ast }}$
provided that
$(X,\Vert \cdot \Vert )$
is a reflexive Banach space.
Theorem 5.2. Let
$X$
be a locally convex space. Let
$\unicode[STIX]{x1D711}$
,
$\unicode[STIX]{x1D713}:X\rightarrow \overline{\mathbb{R}}$
be functions such that
$\unicode[STIX]{x1D713}\in {\mathcal{E}}^{\unicode[STIX]{x1D711}}$
and
$\unicode[STIX]{x1D711}\in {\mathcal{E}}^{\unicode[STIX]{x1D713}}$
. Assume that either:
-
– the space
$X$
is finite-dimensional; or -
– one of the functions
$(-\unicode[STIX]{x1D711})^{\ast }$
and
$(-\unicode[STIX]{x1D713})^{\ast }$
is
$\unicode[STIX]{x1D70F}(X^{\ast },X)$
-continuous at some point and finite at this point.
Then we have
$(-\unicode[STIX]{x1D711})^{\ast \ast }\sim (-\unicode[STIX]{x1D713})^{\ast \ast }$
. If each of the functions
$-\unicode[STIX]{x1D711}$
and
$-\unicode[STIX]{x1D713}$
has a continuous affine minorant, then
$\overline{\text{co}}(-\unicode[STIX]{x1D711})\sim \overline{\text{co}}(-\unicode[STIX]{x1D713})$
. In particular, if
$-\unicode[STIX]{x1D711}\in \unicode[STIX]{x1D6E4}_{0}(X)$
and
$-\unicode[STIX]{x1D713}\in \unicode[STIX]{x1D6E4}_{0}(X)$
, then we have
$\unicode[STIX]{x1D711}\sim \unicode[STIX]{x1D713}$
.
Proof. By assumption, we have
$-\unicode[STIX]{x1D713}=(-\unicode[STIX]{x1D711}){\bigtriangledown}f$
and
$-\unicode[STIX]{x1D711}=(-\unicode[STIX]{x1D713}){\bigtriangledown}g$
, for some
$f$
,
$g:X\rightarrow \overline{\mathbb{R}}$
. Taking the conjugates, we obtain that
First observe that if one of the functions
$(-\unicode[STIX]{x1D711})^{\ast }$
,
$(-\unicode[STIX]{x1D713})^{\ast }$
,
$f^{\ast }$
or
$g^{\ast }$
is equal to
$-\unicode[STIX]{x1D714}_{X^{\ast }}$
, then equalities (36) imply that
$(-\unicode[STIX]{x1D711})^{\ast }=(-\unicode[STIX]{x1D713})^{\ast }=-\unicode[STIX]{x1D714}_{X^{\ast }}$
. This implies in turn that
$\unicode[STIX]{x1D711}=\unicode[STIX]{x1D713}=-\unicode[STIX]{x1D714}_{X}$
and the conclusion is satisfied. From now on, let us assume that the functions
$(-\unicode[STIX]{x1D711})^{\ast }$
,
$(-\unicode[STIX]{x1D713})^{\ast }$
,
$f^{\ast }$
and
$g^{\ast }$
differ from
$-\unicode[STIX]{x1D714}_{X^{\ast }}$
. From the first equality of (36), we deduce that
$\text{dom}(-\unicode[STIX]{x1D713})^{\ast }\subset \text{dom}(-\unicode[STIX]{x1D711})^{\ast }$
, while the second equality of (36) yields
$\text{dom}(-\unicode[STIX]{x1D711})^{\ast }\subset \text{dom}(-\unicode[STIX]{x1D713})^{\ast }$
. Finally, the domains of
$(-\unicode[STIX]{x1D711})^{\ast }$
and
$(-\unicode[STIX]{x1D713})^{\ast }$
coincide and both functions are finite on their common domain
$D$
. If the set
$D$
is empty, then
$(-\unicode[STIX]{x1D711})^{\ast }=(-\unicode[STIX]{x1D713})^{\ast }=\unicode[STIX]{x1D714}_{X^{\ast }}$
. This implies that
$(-\unicode[STIX]{x1D711})^{\ast \ast }=(-\unicode[STIX]{x1D713})^{\ast \ast }=-\unicode[STIX]{x1D714}_{X}$
, hence the conclusion is trivially satisfied. Without loss of generality, we now assume that
$D\neq \emptyset$
. By combining both equalities of (36), we obtain
$$\begin{eqnarray}(-\unicode[STIX]{x1D711})^{\ast }=(-\unicode[STIX]{x1D711})^{\ast }+f^{\ast }+g^{\ast }.\end{eqnarray}$$
It ensues that
$f^{\ast }+g^{\ast }=0$
on
$D$
. Hence, the function
${f^{\ast }}_{|D}$
is finite-valued on
$D$
and both convex and concave. By applying the previous lemma with
$h={f^{\ast }}_{|D}$
, we obtain that there exist a linear function
$\ell :X^{\ast }\rightarrow \mathbb{R}$
and
$\unicode[STIX]{x1D6FC}\in \mathbb{R}$
such that
$f^{\ast }=\ell +\unicode[STIX]{x1D6FC}$
on
$D$
. Coming back to the first equality of (36), we deduce that
$$\begin{eqnarray}(-\unicode[STIX]{x1D713})^{\ast }=(-\unicode[STIX]{x1D711})^{\ast }+\ell +\unicode[STIX]{x1D6FC}.\end{eqnarray}$$
Observe that the above equality holds true on the whole space
$X^{\ast }$
, since the functions
$(-\unicode[STIX]{x1D711})^{\ast }$
and
$(-\unicode[STIX]{x1D713})^{\ast }$
are equal to
$+\infty$
outside
$D$
. Taking the conjugate of each member, we find for every
$\unicode[STIX]{x1D709}\in X$
,
$$\begin{eqnarray}\displaystyle (-\unicode[STIX]{x1D713})^{\ast \ast }(\unicode[STIX]{x1D709})=\sup _{x^{\ast }\in X^{\ast }}[\langle x^{\ast },\unicode[STIX]{x1D709}\rangle -(-\unicode[STIX]{x1D711})^{\ast }(x^{\ast })-\ell (x^{\ast })-\unicode[STIX]{x1D6FC}]. & & \displaystyle\end{eqnarray}$$
Let us now show that the linear function
$\ell$
is
$\unicode[STIX]{x1D70F}(X^{\ast },X)$
-continuous on
$X^{\ast }$
.
Lemma 5.1. Under the assumptions of Theorem 5.2, the function
$\ell :X^{\ast }\rightarrow \mathbb{R}$
is
$\unicode[STIX]{x1D70F}(X^{\ast },X)$
-continuous on
$X^{\ast }$
.
Proof of Lemma 5.1.
If the space
$X$
is finite-dimensional, the assertion is obvious. Now assume that the function
$(-\unicode[STIX]{x1D711})^{\ast }$
is
$\unicode[STIX]{x1D70F}(X^{\ast },X)$
-continuous at some
$\overline{x}^{\ast }\in X^{\ast }$
and finite at this point. There exist a
$\unicode[STIX]{x1D70F}(X^{\ast },X)$
-neighborhood
$W$
of
$\overline{x}^{\ast }$
and
$M\in \mathbb{R}$
such that
$(-\unicode[STIX]{x1D711})^{\ast }\leqslant M$
on
$W$
. We deduce from (37) that for every
$\unicode[STIX]{x1D709}\in X$
,
$$\begin{eqnarray}\displaystyle (-\unicode[STIX]{x1D713})^{\ast \ast }(\unicode[STIX]{x1D709}) & {\geqslant} & \displaystyle \sup _{x^{\ast }\in W}[\langle x^{\ast },\unicode[STIX]{x1D709}\rangle -(-\unicode[STIX]{x1D711})^{\ast }(x^{\ast })-\ell (x^{\ast })-\unicode[STIX]{x1D6FC}]\nonumber\\ \displaystyle & {\geqslant} & \displaystyle \sup _{x^{\ast }\in W}[\langle x^{\ast },\unicode[STIX]{x1D709}\rangle -\ell (x^{\ast })]-M-\unicode[STIX]{x1D6FC}.\nonumber\end{eqnarray}$$
Let us argue by contradiction and assume that
$\ell$
is not
$\unicode[STIX]{x1D70F}(X^{\ast },X)$
-continuous on
$X^{\ast }$
. Since the linear function
$\langle \cdot ,\unicode[STIX]{x1D709}\rangle -\ell$
is not
$\unicode[STIX]{x1D70F}(X^{\ast },X)$
-continuous on
$X^{\ast }$
, the above supremum equals
$+\infty$
. It ensues that
$(-\unicode[STIX]{x1D713})^{\ast \ast }=\unicode[STIX]{x1D714}_{X}$
, and hence
$-\unicode[STIX]{x1D713}=\unicode[STIX]{x1D714}_{X}$
. Recalling that
$-\unicode[STIX]{x1D711}=(-\unicode[STIX]{x1D713}){\bigtriangledown}g$
, we deduce that
$-\unicode[STIX]{x1D711}=\unicode[STIX]{x1D714}_{X}$
. This implies in turn that
$(-\unicode[STIX]{x1D711})^{\ast }=-\unicode[STIX]{x1D714}_{X^{\ast }}$
, a contradiction with
$(-\unicode[STIX]{x1D711})^{\ast }(\overline{x}^{\ast })\in \mathbb{R}$
. We conclude that the linear function
$\ell$
is
$\unicode[STIX]{x1D70F}(X^{\ast },X)$
-continuous on
$X^{\ast }$
. Since
$\unicode[STIX]{x1D711}$
and
$\unicode[STIX]{x1D713}$
play symmetric roles, the same conclusion holds true if the function
$(-\unicode[STIX]{x1D713})^{\ast }$
is assumed to be
$\unicode[STIX]{x1D70F}(X^{\ast },X)$
-continuous at some
$\tilde{x}^{\ast }\in X^{\ast }$
and finite at this point.◻
From the previous lemma and the definition of the Mackey topology
$\unicode[STIX]{x1D70F}(X^{\ast },X)$
, there exists
$x\in X$
such that
$\ell (x^{\ast })=\langle x^{\ast },x\rangle$
for every
$x^{\ast }\in X^{\ast }$
. In view of (37), we deduce that
$$\begin{eqnarray}(-\unicode[STIX]{x1D713})^{\ast \ast }(\unicode[STIX]{x1D709})=\sup _{x^{\ast }\in X^{\ast }}[\langle x^{\ast },\unicode[STIX]{x1D709}-x\rangle -(-\unicode[STIX]{x1D711})^{\ast }(x^{\ast })]-\unicode[STIX]{x1D6FC}=(-\unicode[STIX]{x1D711})^{\ast \ast }(\unicode[STIX]{x1D709}-x)-\unicode[STIX]{x1D6FC}.\end{eqnarray}$$
Since this is true for every
$\unicode[STIX]{x1D709}\in X$
, we conclude that
$(-\unicode[STIX]{x1D713})^{\ast \ast }\sim (-\unicode[STIX]{x1D711})^{\ast \ast }$
. If the function
$(-\unicode[STIX]{x1D711})$
(respectively
$(-\unicode[STIX]{x1D713})$
) admits a continuous affine minorant, we have
$(-\unicode[STIX]{x1D711})^{\ast \ast }=\overline{\text{co}}(-\unicode[STIX]{x1D711})$
(respectively
$(-\unicode[STIX]{x1D713})^{\ast \ast }=\overline{\text{co}}(-\unicode[STIX]{x1D713})$
). We infer that
$\overline{\text{co}}(-\unicode[STIX]{x1D713})\sim \overline{\text{co}}(-\unicode[STIX]{x1D711})$
. The last assertion of the statement is a direct consequence of what precedes.◻
Remark 5.1. If the normed space
$(X,\Vert \cdot \Vert )$
is reflexive, the
$\unicode[STIX]{x1D70F}(X^{\ast },X)$
-continuity assumption on
$(-\unicode[STIX]{x1D711})^{\ast }$
(respectively
$(-\unicode[STIX]{x1D713})^{\ast }$
) amounts to the continuity assumption with respect to the dual norm
$\Vert \cdot \Vert _{X^{\ast }}$
.
Theorem 5.2 implies that the relation
$\preccurlyeq$
defines an order on the following set:
$$\begin{eqnarray}\{\unicode[STIX]{x1D711}\in -\unicode[STIX]{x1D6E4}_{0}(X),(-\unicode[STIX]{x1D711})^{\ast }\text{is }\unicode[STIX]{x1D70F}(X^{\ast },X)\text{-continuous at some point}\}/\sim .\end{eqnarray}$$
If the space
$X$
is finite-dimensional, the relation
$\preccurlyeq$
is an order on the set
$(-\unicode[STIX]{x1D6E4}_{0}(X))/\sim$
.
By applying Theorem 5.2 with the opposite of indicator functions, we obtain the following corollary.
Corollary 5.2. Let
$X$
be a locally convex space. Let
$\unicode[STIX]{x1D6E4}$
,
$\unicode[STIX]{x1D6E5}\subset X$
be such that
$\unicode[STIX]{x1D6E5}\in {\mathcal{I}}^{\unicode[STIX]{x1D6E4}}$
and
$\unicode[STIX]{x1D6E4}\in {\mathcal{I}}^{\unicode[STIX]{x1D6E5}}$
. Assume that either:
-
– the space
$X$
is finite-dimensional; or -
– one of the functions
$\unicode[STIX]{x1D70E}_{X\setminus \unicode[STIX]{x1D6E4}}$
and
$\unicode[STIX]{x1D70E}_{X\setminus \unicode[STIX]{x1D6E5}}$
is
$\unicode[STIX]{x1D70F}(X^{\ast },X)$
-continuous at some point.
Then we have
$\overline{\text{co}}(X\setminus \unicode[STIX]{x1D6E4})\sim \overline{\text{co}}(X\setminus \unicode[STIX]{x1D6E5})$
. In particular, if the sets
$X\setminus \unicode[STIX]{x1D6E4}$
and
$X\setminus \unicode[STIX]{x1D6E5}$
are closed and convex, then
$\unicode[STIX]{x1D6E4}\sim \unicode[STIX]{x1D6E5}$
.
Proof. From Corollary 4.3(i), condition
$\unicode[STIX]{x1D6E5}\in {\mathcal{I}}^{\unicode[STIX]{x1D6E4}}$
(respectively
$\unicode[STIX]{x1D6E4}\in {\mathcal{I}}^{\unicode[STIX]{x1D6E5}}$
) is equivalent to
$-\unicode[STIX]{x1D6FF}_{X\setminus \unicode[STIX]{x1D6E5}}\in {\mathcal{E}}^{-\unicode[STIX]{x1D6FF}_{X\setminus \unicode[STIX]{x1D6E4}}}$
(respectively
$-\unicode[STIX]{x1D6FF}_{X\setminus \unicode[STIX]{x1D6E4}}\in {\mathcal{E}}^{-\unicode[STIX]{x1D6FF}_{X\setminus \unicode[STIX]{x1D6E5}}}$
). By applying Theorem 5.2 with
$\unicode[STIX]{x1D711}=-\unicode[STIX]{x1D6FF}_{X\setminus \unicode[STIX]{x1D6E4}}$
and
$\unicode[STIX]{x1D713}=-\unicode[STIX]{x1D6FF}_{X\setminus \unicode[STIX]{x1D6E5}}$
, we obtain the existence of
$\unicode[STIX]{x1D709}\in X$
and
$\unicode[STIX]{x1D6FC}\in \mathbb{R}$
such that
$$\begin{eqnarray}\overline{\text{co}}(\unicode[STIX]{x1D6FF}_{X\setminus \unicode[STIX]{x1D6E5}})=[\overline{\text{co}}(\unicode[STIX]{x1D6FF}_{X\setminus \unicode[STIX]{x1D6E4}})](\cdot -\unicode[STIX]{x1D709})-\unicode[STIX]{x1D6FC}.\end{eqnarray}$$
We immediately deduce that
$\overline{\text{co}}(X\setminus \unicode[STIX]{x1D6E5})=\overline{\text{co}}(X\setminus \unicode[STIX]{x1D6E4})+\unicode[STIX]{x1D709}$
. The last assertion of the statement is a direct consequence of what precedes.◻
5.3 A counterexample
Let us start with a preliminary result.
Lemma 5.4. Let
$X$
be a topological vector space and let
$G$
be a dense additive subgroup of
$X$
. Assume that
$K\subset X$
is an open set such that
$K+K\subset K$
and
$0\in \text{cl}(K)$
. Then we have:
-
(i) for all
$\unicode[STIX]{x1D709}$
,
$\unicode[STIX]{x1D709}^{\prime }\in X$
,
$$\begin{eqnarray}[G\cap (K+\unicode[STIX]{x1D709})]+[G\cap (K+\unicode[STIX]{x1D709}^{\prime })]=G\cap (K+\unicode[STIX]{x1D709}+\unicode[STIX]{x1D709}^{\prime });\end{eqnarray}$$
-
(ii) if, in addition,
$\text{cl}(K)\cap -\text{cl}(K)=\{0\}$
, then if
$$\begin{eqnarray}G\cap (K+\unicode[STIX]{x1D709})=(G\cap K)+\unicode[STIX]{x1D709}^{\prime }\;\Longrightarrow \;\unicode[STIX]{x1D709}=\unicode[STIX]{x1D709}^{\prime };\end{eqnarray}$$
$G\neq X$
and
$\unicode[STIX]{x1D709}\in X\setminus G$
, there is no
$\unicode[STIX]{x1D709}^{\prime }\in X$
such that
$G\cap (K+\unicode[STIX]{x1D709})=(G\cap K)+\unicode[STIX]{x1D709}^{\prime }$
.
Proof. (i) Let us fix
$\unicode[STIX]{x1D709},\unicode[STIX]{x1D709}^{\prime }\in X$
and let us prove the inclusion from the left to the right. Observe that
$$\begin{eqnarray}[G\cap (K+\unicode[STIX]{x1D709})]+[G\cap (K+\unicode[STIX]{x1D709}^{\prime })]\subset G+G\end{eqnarray}$$
and
$$\begin{eqnarray}[G\cap (K+\unicode[STIX]{x1D709})]+[G\cap (K+\unicode[STIX]{x1D709}^{\prime })]\subset (K+\unicode[STIX]{x1D709})+(K+\unicode[STIX]{x1D709}^{\prime }).\end{eqnarray}$$
Since
$G+G\subset G$
and
$K+K\subset K$
, we deduce that
$$\begin{eqnarray}[G\cap (K+\unicode[STIX]{x1D709})]+[G\cap (K+\unicode[STIX]{x1D709}^{\prime })]\subset G\cap (K+\unicode[STIX]{x1D709}+\unicode[STIX]{x1D709}^{\prime }).\end{eqnarray}$$
Now let us establish the reverse inclusion. Let
$x\in G\cap (K+\unicode[STIX]{x1D709}+\unicode[STIX]{x1D709}^{\prime })$
. Observe that the open set
$K+\unicode[STIX]{x1D709}+\unicode[STIX]{x1D709}^{\prime }-x$
contains
$0$
. Recalling that
$0\in \text{cl}(K)$
, we have
$(K+\unicode[STIX]{x1D709}+\unicode[STIX]{x1D709}^{\prime }-x)\cap -K\neq \emptyset$
, hence
$(K+\unicode[STIX]{x1D709}-x)\cap (-K-\unicode[STIX]{x1D709}^{\prime })\neq \emptyset$
. Since the set
$K$
is open, the set
$(K+\unicode[STIX]{x1D709}-x)\cap (-K-\unicode[STIX]{x1D709}^{\prime })$
is open. By using the density of
$G$
in
$X$
, we deduce that
$$\begin{eqnarray}G\cap (K+\unicode[STIX]{x1D709}-x)\cap (-K-\unicode[STIX]{x1D709}^{\prime })\neq \emptyset .\end{eqnarray}$$
Since
$G=-G$
, the above property can be rewritten as
$$\begin{eqnarray}[G\cap (K+\unicode[STIX]{x1D709}-x)]\cap [-G\cap (-K-\unicode[STIX]{x1D709}^{\prime })]\neq \emptyset ,\end{eqnarray}$$
which is, in turn, equivalent to
$$\begin{eqnarray}0\in [G\cap (K+\unicode[STIX]{x1D709}-x)]+[G\cap (K+\unicode[STIX]{x1D709}^{\prime })].\end{eqnarray}$$
Recalling that
$x\in G$
, we have
$G=G-x$
, hence
$$\begin{eqnarray}G\cap (K+\unicode[STIX]{x1D709}-x)=[G\cap (K+\unicode[STIX]{x1D709})]-x.\end{eqnarray}$$
In view of the latter inclusion, we conclude that
$$\begin{eqnarray}x\in [G\cap (K+\unicode[STIX]{x1D709})]+[G\cap (K+\unicode[STIX]{x1D709}^{\prime })].\end{eqnarray}$$
The inclusion
$$\begin{eqnarray}G\cap (K+\unicode[STIX]{x1D709}+\unicode[STIX]{x1D709}^{\prime })\subset [G\cap (K+\unicode[STIX]{x1D709})]+[G\cap (K+\unicode[STIX]{x1D709}^{\prime })]\end{eqnarray}$$
is proved.
(ii) Let us assume that
$G\cap (K+\unicode[STIX]{x1D709})=(G\cap K)+\unicode[STIX]{x1D709}^{\prime }$
for some
$\unicode[STIX]{x1D709},\unicode[STIX]{x1D709}^{\prime }\in X$
. We deduce that
$G\cap (K+\unicode[STIX]{x1D709})\subset K+\unicode[STIX]{x1D709}^{\prime }$
. By using the openness of the set
$K+\unicode[STIX]{x1D709}$
along with the density of
$G$
in
$X$
, we easily infer that
$K+\unicode[STIX]{x1D709}\subset \text{cl}(K)+\unicode[STIX]{x1D709}^{\prime }$
. This implies, in turn, that
$\text{cl}(K)+\unicode[STIX]{x1D709}\subset \text{cl}(K)+\unicode[STIX]{x1D709}^{\prime }$
and since
$0\in \text{cl}(K)$
, we obtain
$\unicode[STIX]{x1D709}-\unicode[STIX]{x1D709}^{\prime }\in \text{cl}(K)$
. By a symmetric argument, we find
$\unicode[STIX]{x1D709}^{\prime }-\unicode[STIX]{x1D709}\in \text{cl}(K)$
, hence
$\unicode[STIX]{x1D709}-\unicode[STIX]{x1D709}^{\prime }\in \text{cl}(K)\cap -\text{cl}(K)$
. Since
$\text{cl}(K)\cap -\text{cl}(K)=\{0\}$
by assumption, we conclude that
$\unicode[STIX]{x1D709}=\unicode[STIX]{x1D709}^{\prime }$
.
Now let
$\unicode[STIX]{x1D709}\in X\setminus G$
and assume that there exists
$\unicode[STIX]{x1D709}^{\prime }\in X$
such that
$G\cap (K+\unicode[STIX]{x1D709})=(G\cap K)+\unicode[STIX]{x1D709}^{\prime }$
. From what precedes, we have
$\unicode[STIX]{x1D709}^{\prime }=\unicode[STIX]{x1D709}$
and, hence,
$G\cap (K+\unicode[STIX]{x1D709})=(G+\unicode[STIX]{x1D709})\cap (K+\unicode[STIX]{x1D709})$
. On the other hand, the assumption
$\unicode[STIX]{x1D709}\in X\setminus G$
implies that the sets
$G$
and
$G+\unicode[STIX]{x1D709}$
are disjoint. We deduce that
$G\cap (K+\unicode[STIX]{x1D709})=\emptyset$
, a contradiction since the non-empty set
$K$
is open and the set
$G$
is dense in
$X$
.◻
Let us now build an example of sets
$\unicode[STIX]{x1D6E4}$
,
$\unicode[STIX]{x1D6E5}\subset X$
satisfying
$\unicode[STIX]{x1D6E5}\in {\mathcal{I}}^{\unicode[STIX]{x1D6E4}}$
and
$\unicode[STIX]{x1D6E4}\in {\mathcal{I}}^{\unicode[STIX]{x1D6E5}}$
, but with
$\unicode[STIX]{x1D6E5}$
and
$\unicode[STIX]{x1D6E4}$
not equal up to a translation. We are given an open set
$K\subset X$
such that
$K+K\subset K$
and
$\text{cl}(K)\cap -\text{cl}(K)=\{0\}$
, along with a dense additive subgroup
$G\subset X$
such that
$G\neq X$
. Define the sets
$C$
,
$U$
,
$V\subset X$
respectively by
$$\begin{eqnarray}C=G\cap K;\qquad U=G\cap (K+\unicode[STIX]{x1D709});\qquad V=G\cap (K-\unicode[STIX]{x1D709}),\end{eqnarray}$$
where
$\unicode[STIX]{x1D709}\in X\setminus G$
. In view of Lemma 5.4(i), the set
$D=C+U$
satisfies
$$\begin{eqnarray}D=G\cap (K+\unicode[STIX]{x1D709})\qquad \text{and}\qquad D+V=G\cap K=C.\end{eqnarray}$$
Lemma 5.4(ii) shows that the set
$D$
is not translated from
$C$
. Defining the complementary sets
$\unicode[STIX]{x1D6E4}=X\setminus C$
and
$\unicode[STIX]{x1D6E5}=X\setminus D$
, we have
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6E5}=X\setminus (C+U)=U^{X\setminus C}=U^{\unicode[STIX]{x1D6E4}}\in {\mathcal{I}}^{\unicode[STIX]{x1D6E4}} & & \displaystyle\end{eqnarray}$$
and
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6E4}=X\setminus (D+V)=V^{X\setminus D}=V^{\unicode[STIX]{x1D6E5}}\in {\mathcal{I}}^{\unicode[STIX]{x1D6E5}}. & & \displaystyle\end{eqnarray}$$
From what precedes, the set
$\unicode[STIX]{x1D6E5}$
is not translated from
$\unicode[STIX]{x1D6E4}$
. The above counterexample for sets obviously furnishes a counterexample for functions. Indeed, we deduce from (38)–(39) that the indicator functions
$\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D6E4}}$
and
$\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D6E5}}$
satisfy
$\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D6E5}}\in {\mathcal{E}}^{\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D6E4}}}$
and
$\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D6E4}}\in {\mathcal{E}}^{\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D6E5}}}$
, but the functions
$\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D6E4}}$
and
$\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D6E5}}$
are not equal up to a translation.
By particularizing the above sets
$G$
,
$K\subset X$
, one obtains various counterexamples. If
$X=\mathbb{R}$
, one can take
$G=\mathbb{Q}$
,
$K=\, ]\!0,+\infty \
$\unicode[STIX]{x1D709}\in \mathbb{R}\setminus \mathbb{Q}$
. On the other hand, if
$X$
is infinite-dimensional, one can assume that
$G$
is a dense subspace of
$X$
and that
$K$
is an open convex cone such that
$\text{cl}(K)$
is pointed. This furnishes a counterexample with convex sets
$C$
,
$D\subset X$
.
6 Cases of either superadditivity or subadditivity of
$\unicode[STIX]{x1D711}$
Let us first recall that a function
$\unicode[STIX]{x1D711}:X\rightarrow \overline{\mathbb{R}}$
is said to be superadditive (respectively subadditive) if for all
$x,y\in X$
,
Let us start with a preliminary result.
Lemma 6.1. Let
$X$
be a vector space. Let
$h,k:X\rightarrow \overline{\mathbb{R}}$
and assume that
$k(0)=0$
. Then we have
As a consequence, the function
$k$
is superadditive if and only if
$k=k\triangle k$
, which is, in turn, equivalent to
$k=k{\bigtriangledown}(-k_{-})$
.
Proof. If
$h=h\triangle k$
, then the definition of
$h\triangle k$
entails that 
for all
$x,y\in X$
. Conversely, if this inequality holds true for every
$x,y\in X$
, we have
for every
$x\in X$
. This implies that
$h(x)=(h\triangle k)(x)$
for every
$x\in X$
and the first equivalence is proved.
For the second equivalence, observe that for all
$x,y\in X$
,
The proof of the third equivalence follows the same lines as the first one. For the last assertion, observe that
$k$
is superadditive if and only if 
for all
$x,y\in X$
. It suffices then to use what precedes with
$h=k$
.◻
Through the above lemma, the following theorem provides, in particular, various characterizations of the class
${\mathcal{E}}^{\unicode[STIX]{x1D711}}$
when
$\unicode[STIX]{x1D711}$
is superadditive.
Theorem 6.1. Let
$X$
be a vector space. Let
$\unicode[STIX]{x1D711}:X\rightarrow \overline{\mathbb{R}}$
be a superadditive function satisfying
$\unicode[STIX]{x1D711}(0)=0$
.
-
(a) For a function
$g:X\rightarrow \overline{\mathbb{R}}$
, the following assertions are equivalent:-
(i)
$g\in {\mathcal{E}}^{\unicode[STIX]{x1D711}}$
; -
(ii)
$g=g\triangle \unicode[STIX]{x1D711}$
; -
(iii)
for all
$x,y\in X$
; -
(iv)
for all
$x,y\in X$
; -
(v)
$g=g{\bigtriangledown}(-\unicode[STIX]{x1D711}_{-})$
; -
(vi)
$-g\in {\mathcal{E}}^{\unicode[STIX]{x1D711}_{-}}$
.
-
-
(b) For every function
$f:X\rightarrow \overline{\mathbb{R}}$
,
$f{\bigtriangledown}(-\unicode[STIX]{x1D711}_{-})$
is the greatest
$\unicode[STIX]{x1D711}$
-envelope that is majorized by
$f$
, while
$f\triangle \unicode[STIX]{x1D711}$
is the lowest
$\unicode[STIX]{x1D711}$
-envelope that is minorized by
$f$
. -
(c) The following inclusion holds true:
${\mathcal{E}}^{-\unicode[STIX]{x1D711}}\subset {\mathcal{E}}^{\unicode[STIX]{x1D711}_{-}}$
.
for all 
for all 
Proof. (a) Let us assume that
$g\in {\mathcal{E}}^{\unicode[STIX]{x1D711}}$
. Then there exists
$f:X\rightarrow \overline{\mathbb{R}}$
such that
$g=f^{\unicode[STIX]{x1D711}}=(-f)\triangle \unicode[STIX]{x1D711}$
. Using the superadditivity of
$\unicode[STIX]{x1D711}$
and the last assertion of Lemma 6.1, we have
$$\begin{eqnarray}g\triangle \unicode[STIX]{x1D711}=((-f)\triangle \unicode[STIX]{x1D711})\triangle \unicode[STIX]{x1D711}=(-f)\triangle (\unicode[STIX]{x1D711}\triangle \unicode[STIX]{x1D711})=(-f)\triangle \unicode[STIX]{x1D711}=g.\end{eqnarray}$$
This shows that (i)
$\Longrightarrow$
(ii). Conversely, if
$g=g\triangle \unicode[STIX]{x1D711}$
then
$g=(-g)^{\unicode[STIX]{x1D711}}$
and clearly
$g\in {\mathcal{E}}^{\unicode[STIX]{x1D711}}$
. The equivalences
$(\text{ii})\Longleftrightarrow (\text{iii})\Longleftrightarrow (\text{iv})\Longleftrightarrow (\text{v})$
follow directly from Lemma 6.1. For the equivalence (v)
$\Longleftrightarrow$
(vi), observe that
$$\begin{eqnarray}g=g{\bigtriangledown}(-\unicode[STIX]{x1D711}_{-})\;\Longleftrightarrow \;-g=(-g)\triangle \unicode[STIX]{x1D711}_{-},\end{eqnarray}$$
and invoke the equivalence (i)
$\Longleftrightarrow$
(ii).
(b) Let
$f:X\rightarrow \overline{\mathbb{R}}$
and take
$g=f{\bigtriangledown}(-\unicode[STIX]{x1D711}_{-})=-f^{\unicode[STIX]{x1D711}_{-}}$
. By using the implication (vi)
$\Longrightarrow$
(ii) in (a), we obtain
$g=g\triangle \unicode[STIX]{x1D711}$
, thus implying that
$g=(f^{\unicode[STIX]{x1D711}_{-}})^{\unicode[STIX]{x1D711}}$
. Hence,
$f{\bigtriangledown}(-\unicode[STIX]{x1D711}_{-})$
coincides with
$(f^{\unicode[STIX]{x1D711}_{-}})^{\unicode[STIX]{x1D711}}$
, which is by property (6) the greatest element of
${\mathcal{E}}^{\unicode[STIX]{x1D711}}$
that is majorized by
$f$
. Replacing
$f$
(respectively
$\unicode[STIX]{x1D711}$
) with
$-f$
(respectively
$\unicode[STIX]{x1D711}_{-}$
) and taking the opposite, we deduce that
$f\triangle \unicode[STIX]{x1D711}$
is the lowest element of
$-{\mathcal{E}}^{\unicode[STIX]{x1D711}_{-}}$
that is minorized by
$f$
. It suffices then to recall that
${\mathcal{E}}^{\unicode[STIX]{x1D711}_{-}}=-{\mathcal{E}}^{\unicode[STIX]{x1D711}}$
, see the equivalence
$(\text{i})\Longleftrightarrow (\text{vi})$
in (a).
(c) Since
$\unicode[STIX]{x1D711}\in {\mathcal{E}}^{\unicode[STIX]{x1D711}}$
, we have
$-\unicode[STIX]{x1D711}\in -{\mathcal{E}}^{\unicode[STIX]{x1D711}}={\mathcal{E}}^{\unicode[STIX]{x1D711}_{-}}$
. In view of Proposition 3.2(iii), we infer that
${\mathcal{E}}^{-\unicode[STIX]{x1D711}}\subset {\mathcal{E}}^{\unicode[STIX]{x1D711}_{-}}$
.◻
Example 6.1. Assume that
$(X,\Vert \cdot \Vert )$
is a normed space. For
$k\geqslant 0$
and
$\unicode[STIX]{x1D6FC}\in \,]\!0,1\!]$
, take
$\unicode[STIX]{x1D711}=-k\Vert \cdot \Vert ^{\unicode[STIX]{x1D6FC}}$
. Observe that for all
$x,y\in X$
,
$$\begin{eqnarray}\displaystyle \Vert x+y\Vert ^{\unicode[STIX]{x1D6FC}}\leqslant (\Vert x\Vert +\Vert y\Vert )^{\unicode[STIX]{x1D6FC}}\leqslant \Vert x\Vert ^{\unicode[STIX]{x1D6FC}}+\Vert y\Vert ^{\unicode[STIX]{x1D6FC}}. & & \displaystyle\end{eqnarray}$$
It ensues that the function
$\Vert \cdot \Vert ^{\unicode[STIX]{x1D6FC}}$
is subadditive, hence
$\unicode[STIX]{x1D711}$
is superadditive. From Theorem 6.1(a), we deduce that
$$\begin{eqnarray}\displaystyle f\in {\mathcal{E}}^{-k\Vert \cdot \Vert ^{\unicode[STIX]{x1D6FC}}}\;\Longleftrightarrow \;f(x)\geqslant f(y)-k\Vert x-y\Vert ^{\unicode[STIX]{x1D6FC}}\quad \text{for all }x,y\in X. & & \displaystyle\end{eqnarray}$$
By reversing the roles of
$x$
and
$y$
, we immediately obtain
$$\begin{eqnarray}\displaystyle f\in {\mathcal{E}}^{-k\Vert \cdot \Vert ^{\unicode[STIX]{x1D6FC}}}\;\Longleftrightarrow \;f(x)\leqslant f(y)+k\Vert x-y\Vert ^{\unicode[STIX]{x1D6FC}}\quad \text{for all }x,y\in X. & & \displaystyle\end{eqnarray}$$
If
$f(y)=+\infty$
(respectively
$f(y)=-\infty$
) for some
$y\in X$
, we deduce from (41) (respectively (42)) that
$f=\unicode[STIX]{x1D714}_{X}$
(respectively
$f=-\unicode[STIX]{x1D714}_{X}$
). On the other hand, if the function
$f$
is finite-valued, we infer from (41)–(42) that
$|f(x)-f(y)|\leqslant k\Vert x-y\Vert ^{\unicode[STIX]{x1D6FC}}$
for all
$x,y\in X$
. This implies that
From Theorem 6.1(b), we deduce that
$f{\bigtriangledown}k\Vert \cdot \Vert ^{\unicode[STIX]{x1D6FC}}$
(respectively
$f\triangle (-k\Vert \cdot \Vert ^{\unicode[STIX]{x1D6FC}})$
) is the greatest (respectively lowest)
$\unicode[STIX]{x1D711}$
-envelope that is majorized (respectively minorized) by
$f$
. Since the map
$\Vert \cdot \Vert ^{\unicode[STIX]{x1D6FC}}$
is even, Theorem 6.1(c) shows that
${\mathcal{E}}^{k\Vert \cdot \Vert ^{\unicode[STIX]{x1D6FC}}}\subset {\mathcal{E}}^{-k\Vert \cdot \Vert ^{\unicode[STIX]{x1D6FC}}}$
.
Now assume that
$\unicode[STIX]{x1D6FC}=1$
. From what precedes, we obtain that
$$\begin{eqnarray}{\mathcal{E}}^{-k\Vert \cdot \Vert }=\{f:X\rightarrow \mathbb{R},f\text{ is }k\text{-Lipschitz continuous}\}\cup \{\unicode[STIX]{x1D714}_{X},-\unicode[STIX]{x1D714}_{X}\}.\end{eqnarray}$$
The Pasch–Hausdorff regularization of
$f$
, defined by
$l_{k}(f)=f{\bigtriangledown}k\Vert \cdot \Vert$
, is the greatest function of
${\mathcal{E}}^{-k\Vert \cdot \Vert }$
that is majorized by
$f$
. On the other hand,
$f\triangle (-k\Vert \cdot \Vert )$
is the lowest function of
${\mathcal{E}}^{-k\Vert \cdot \Vert }$
that is minorized by
$f$
. The inclusion
${\mathcal{E}}^{k\Vert \cdot \Vert }\subset {\mathcal{E}}^{-k\Vert \cdot \Vert }$
shows that the
$k\Vert \cdot \Vert$
-envelopes are either
$k$
-Lipschitz continuous or equal to
$\pm \unicode[STIX]{x1D714}_{X}$
. The convexity of
$\Vert \cdot \Vert$
implies that
$k\Vert \cdot \Vert$
-envelopes are also convex, therefore the inclusion
${\mathcal{E}}^{k\Vert \cdot \Vert }\subset {\mathcal{E}}^{-k\Vert \cdot \Vert }$
is strict. This ensures that the inclusion in (c) of the above theorem generally fails to be an equality.
As regards the function
$\unicode[STIX]{x1D711}=-k\Vert \cdot \Vert ^{\unicode[STIX]{x1D6FC}}$
it is also worth mentioning that, for
$\unicode[STIX]{x1D702}(x,y):=\Vert x-y\Vert ^{\unicode[STIX]{x1D6FC}}$
with
$\unicode[STIX]{x1D6FC}>0$
(even with more general coupling functions) and taking
$$\begin{eqnarray}\boldsymbol{\unicode[STIX]{x1D6F7}}:=\{r-\unicode[STIX]{x1D70E}\unicode[STIX]{x1D702}(\cdot ,y):r\in \mathbb{R},\unicode[STIX]{x1D70E}>0,y\in X\},\end{eqnarray}$$
a lower semicontinuous function on the normed space
$X$
is shown in [Reference Dolecki and Kurcyusz7, Theorem 4.2] to be
$\boldsymbol{\unicode[STIX]{x1D6F7}}$
-convex (i.e. a pointwise supremum of functions in
$\boldsymbol{\unicode[STIX]{x1D6F7}}$
), whenever it is bounded from below by a function in
$\boldsymbol{\unicode[STIX]{x1D6F7}}$
. The latter property with
$\unicode[STIX]{x1D6FC}=2$
was previously proved in [Reference Rockafellar29, Theorem 2]. The function
$(x,y)\mapsto -k\Vert x-y\Vert ^{\unicode[STIX]{x1D6FC}}$
is also used as a particular important example of coupling functions arising in the framework of generalized conjugacy in many papers, see, for example, [Reference Penot, Volle and Cambini24, p. 204].
Remark 6.1. Given a non-increasing convex function
$\unicode[STIX]{x1D703}:\mathbb{R}_{+}\rightarrow \mathbb{R}$
such that
$\unicode[STIX]{x1D703}(0)=0$
, one can easily check that the function
$\unicode[STIX]{x1D703}(\Vert \cdot \Vert )$
is superadditive. Hence, the previous example can be generalized by taking
$\unicode[STIX]{x1D711}=\unicode[STIX]{x1D703}(\Vert \cdot \Vert )$
.
Example 6.2. Let
$X$
be a vector space. Let
$\unicode[STIX]{x1D6EC}\subset X$
be a set containing the origin and such that
$\unicode[STIX]{x1D6EC}+\unicode[STIX]{x1D6EC}\subset \unicode[STIX]{x1D6EC}$
. The function
$\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D6EC}}$
is clearly subadditive. This implies that the function
$\unicode[STIX]{x1D711}=-\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D6EC}}$
is superadditive. By Theorem 6.1(a) it follows that
This and Theorem 6.1(b) entail that
$f{\bigtriangledown}\unicode[STIX]{x1D6FF}_{-\unicode[STIX]{x1D6EC}}$
(respectively
$f\triangle (-\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D6EC}})$
) is the greatest (respectively lowest)
$\unicode[STIX]{x1D6EC}$
-non-decreasing function that is majorized (respectively minorized) by
$f$
. Further, Theorem 6.1(c) says that
${\mathcal{E}}^{\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D6EC}}}\subset {\mathcal{E}}^{(-\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D6EC}})_{-}}={\mathcal{E}}^{-\unicode[STIX]{x1D6FF}_{-\unicode[STIX]{x1D6EC}}}$
, hence the functions of
${\mathcal{E}}^{\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D6EC}}}$
are
$\unicode[STIX]{x1D6EC}$
-non-increasing. In fact, this can be recovered directly by using the characterization of
${\mathcal{E}}^{\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D6EC}}}$
given by Proposition 4.4(ii).
7 Case
$\unicode[STIX]{x1D711}\in \unicode[STIX]{x1D6E4}(X)$
7.1 Expressions of
$\unicode[STIX]{x1D711}$
-envelopes as Legendre–Fenchel conjugates
Let us start with the following elementary lemma.
Lemma 7.1. Let
$X$
be a locally convex space. For every function
$f:X\rightarrow \overline{\mathbb{R}}$
, we have
$(f^{\ast })_{-}=(f_{-})^{\ast }$
.
Proof. It suffices to use the definition of the Legendre–Fenchel conjugate. For every
$\unicode[STIX]{x1D709}^{\ast }\in X^{\ast }$
, we have
$$\begin{eqnarray}\displaystyle \hspace{30.00005pt}(f^{\ast })_{-}(\unicode[STIX]{x1D709}^{\ast })=(f^{\ast })(-\unicode[STIX]{x1D709}^{\ast }) & = & \displaystyle \sup _{x\in X}\{\langle -\unicode[STIX]{x1D709}^{\ast },x\rangle -f(x)\}\nonumber\\ \displaystyle & = & \displaystyle \sup _{y\in X}\{\langle \unicode[STIX]{x1D709}^{\ast },y\rangle -f(-y)\}\nonumber\\ \displaystyle & = & \displaystyle \sup _{y\in X}\{\langle \unicode[STIX]{x1D709}^{\ast },y\rangle -f_{-}(y)\}=(f_{-})^{\ast }(\unicode[STIX]{x1D709}^{\ast }).\hspace{13.99995pt}\square\nonumber\end{eqnarray}$$
Theorem 7.1. Let
$X$
be a locally convex space. Let us assume that
$\unicode[STIX]{x1D711}\in \unicode[STIX]{x1D6E4}(X)$
and let
$\unicode[STIX]{x1D713}:X^{\ast }\rightarrow \overline{\mathbb{R}}$
be such that
$\unicode[STIX]{x1D713}^{\ast }=\unicode[STIX]{x1D711}$
. Then we have for every function
$f:X\rightarrow \overline{\mathbb{R}}$
,
Moreover, the following equivalences hold:
Proof. For every
$x\in X$
,
For the first equivalence, recall that
$g\in {\mathcal{E}}^{\unicode[STIX]{x1D711}}$
if and only if there exists
$f:X\rightarrow \overline{\mathbb{R}}$
such that
$g=f^{\unicode[STIX]{x1D711}}$
. Then use the equality 
and the fact that the range of the Legendre–Fenchel transform is equal to
$\unicode[STIX]{x1D6E4}(X^{\ast })$
, see, for example, [Reference Moreau22].
For the second equivalence, observe that
Remark 7.1. Since
$\unicode[STIX]{x1D711}\in \unicode[STIX]{x1D6E4}(X)$
by assumption, we have
$\unicode[STIX]{x1D711}^{\ast \ast }=\unicode[STIX]{x1D711}$
, hence we can take
$\unicode[STIX]{x1D713}=\unicode[STIX]{x1D711}^{\ast }$
in the statement of Theorem 7.1.
Remark 7.2. Formula (43) can be recovered partially by using a formula on the conjugate of the difference of functions. Recall that for
$\unicode[STIX]{x1D713}:X\rightarrow \mathbb{R}\cup \{+\infty \}$
and
$h\in \unicode[STIX]{x1D6E4}_{0}(X)$
,
This formula is due to Hiriart-Urruty [Reference Hiriart-Urruty11]. It was established first by Pshenichnyi [Reference Pshenichnyi27], assuming that both
$\unicode[STIX]{x1D713}$
and
$h$
are finite-valued convex functions. Now let
$\unicode[STIX]{x1D711}\in \unicode[STIX]{x1D6E4}_{0}(X)$
and
$f\in \unicode[STIX]{x1D6E4}_{0}(X)$
. By reversing the roles of
$X$
and
$X^{\ast }$
and by using equality (44) with
$h=(f_{-})^{\ast }$
and
$\unicode[STIX]{x1D713}:X^{\ast }\rightarrow \mathbb{R}\cup \{+\infty \}$
such that
$\unicode[STIX]{x1D713}^{\ast }=\unicode[STIX]{x1D711}$
, we find
Hence, we recover formula (43) in the case where both functions
$\unicode[STIX]{x1D711}$
and
$f$
are in
$\unicode[STIX]{x1D6E4}_{0}(X)$
.
The next corollary says, in particular, that the
$\unicode[STIX]{x1D711}$
-envelope of a function coincides with the
$\unicode[STIX]{x1D711}$
-envelope of its lower semicontinuous convex hull whenever
$\unicode[STIX]{x1D711}\in \unicode[STIX]{x1D6E4}(X)$
.
Corollary 7.1. Let
$X$
be a locally convex space and
$\unicode[STIX]{x1D711}\in \unicode[STIX]{x1D6E4}(X)$
. Then we have for every function
$f:X\rightarrow \overline{\mathbb{R}}$
and every function
$g:X\rightarrow \overline{\mathbb{R}}$
satisfying
$\overline{\text{co}}\,f\leqslant g\leqslant f$
,
$$\begin{eqnarray}f^{\unicode[STIX]{x1D711}}=(\overline{\text{co}}\,f)^{\unicode[STIX]{x1D711}}=g^{\unicode[STIX]{x1D711}}.\end{eqnarray}$$
Proof. For the first equality, it suffices to use Theorem 7.1 and the fact that the functions
$f$
and
$\overline{\text{co}}\,f$
have the same Legendre–Fenchel conjugate. On the other hand, since
$\overline{\text{co}}\,f\leqslant g\leqslant f$
, we see that
$f^{\unicode[STIX]{x1D711}}\leqslant g^{\unicode[STIX]{x1D711}}\leqslant (\overline{\text{co}}\,f)^{\unicode[STIX]{x1D711}}$
. Recalling that
$f^{\unicode[STIX]{x1D711}}=(\overline{\text{co}}\,f)^{\unicode[STIX]{x1D711}}$
, the second equality immediately follows.◻
For every set
$D\subset X^{\ast }$
, we define as in §3 the classes
$\unicode[STIX]{x1D6F4}_{D}$
and
$\unicode[STIX]{x1D6F4}_{D}^{\ast }$
by
$$\begin{eqnarray}\unicode[STIX]{x1D6F4}_{D}=\{f:X^{\ast }\rightarrow \overline{\mathbb{R}},\text{dom}\,f\subset D\}\qquad \text{and}\qquad \unicode[STIX]{x1D6F4}_{D}^{\ast }=\{f^{\ast },f\in \unicode[STIX]{x1D6F4}_{D}\}.\end{eqnarray}$$
In the same vein, let us define the classes
$\widehat{\unicode[STIX]{x1D6F4}}_{D}$
and
$\widehat{\unicode[STIX]{x1D6F4}}_{D}^{\ast }$
by
$$\begin{eqnarray}\widehat{\unicode[STIX]{x1D6F4}}_{D}=\{f:X^{\ast }\rightarrow \overline{\mathbb{R}},\text{dom}\,f=D\}\qquad \text{and}\qquad \widehat{\unicode[STIX]{x1D6F4}}_{D}^{\ast }=\{f^{\ast },f\in \widehat{\unicode[STIX]{x1D6F4}}_{D}\}.\end{eqnarray}$$
The following proposition allows us to characterize the classes
$\widehat{\unicode[STIX]{x1D6F4}}_{D}^{\ast }$
and
$\unicode[STIX]{x1D6F4}_{D}^{\ast }$
.
Proposition 7.1. Let
$X$
be a locally convex space and let
$D\subset X^{\ast }$
be such that
$D=\{a_{i}^{\ast },i\in I\}$
for some set
$I$
. Then for every function
$f:X\rightarrow \overline{\mathbb{R}}$
, we have
$f\in \widehat{\unicode[STIX]{x1D6F4}}_{D}^{\ast }$
(respectively
$\unicode[STIX]{x1D6F4}_{D}^{\ast }$
) if and only if there exists a family
$(\unicode[STIX]{x1D6FC}_{i})_{i\in I}\subset \mathbb{R}\cup \{+\infty \}$
(respectively
$\overline{\mathbb{R}}$
) such that
$f=\sup _{i\in I}\langle a_{i}^{\ast },\cdot \rangle +~\unicode[STIX]{x1D6FC}_{i}$
.
Proof. Assume that
$f\in \widehat{\unicode[STIX]{x1D6F4}}_{D}^{\ast }$
(respectively
$\unicode[STIX]{x1D6F4}_{D}^{\ast }$
). By definition, there exists
$g:X^{\ast }\rightarrow \overline{\mathbb{R}}$
such that
$f=g^{\ast }$
and
$\text{dom}\,g=D$
(respectively
$\text{dom}\,g\subset D$
). Hence, we have
$$\begin{eqnarray}f=\sup _{x^{\ast }\in D}\langle x^{\ast },\cdot \rangle -g(x^{\ast })=\sup _{i\in I}\langle a_{i}^{\ast },\cdot \rangle -g(a_{i}^{\ast }).\end{eqnarray}$$
By setting
$\unicode[STIX]{x1D6FC}_{i}=-g(a_{i}^{\ast })$
for every
$i\in I$
, we obtain
$f=\sup _{i\in I}\langle a_{i}^{\ast },\cdot \rangle +\unicode[STIX]{x1D6FC}_{i}$
with
$\unicode[STIX]{x1D6FC}_{i}\in \mathbb{R}\cup \{+\infty \}$
(respectively
$\overline{\mathbb{R}}$
).
Conversely, assume that there exists
$(\unicode[STIX]{x1D6FC}_{i})_{i\in I}\subset \mathbb{R}\cup \{+\infty \}$
(respectively
$\overline{\mathbb{R}}$
) such that
$f=\sup _{i\in I}\langle a_{i}^{\ast },\cdot \rangle +~\unicode[STIX]{x1D6FC}_{i}$
. Then we have
$$\begin{eqnarray}\displaystyle f & = & \displaystyle \sup _{x^{\ast }\in D}\Bigl[\sup _{\{i\in I,a_{i}^{\ast }=x^{\ast }\}}\langle a_{i}^{\ast },\cdot \rangle +\unicode[STIX]{x1D6FC}_{i}\Bigr]\nonumber\\ \displaystyle & = & \displaystyle \sup _{x^{\ast }\in D}\Bigl[\langle x^{\ast },\cdot \rangle +\sup _{\{i\in I,a_{i}^{\ast }=x^{\ast }\}}\unicode[STIX]{x1D6FC}_{i}\Bigr].\nonumber\end{eqnarray}$$
Defining the function
$h:X^{\ast }\rightarrow \overline{\mathbb{R}}$
by
$$\begin{eqnarray}h(x^{\ast })=\left\{\begin{array}{@{}ll@{}}\displaystyle \sup _{\{i\in I,a_{i}^{\ast }=x^{\ast }\}}\unicode[STIX]{x1D6FC}_{i} & \text{if }x^{\ast }\in D,\\ \displaystyle -\infty & \text{if }x^{\ast }\notin D,\end{array}\right.\end{eqnarray}$$
we obtain
$$\begin{eqnarray}\displaystyle f & = & \displaystyle \sup _{x^{\ast }\in D}\langle x^{\ast },\cdot \rangle +h(x^{\ast })\nonumber\\ \displaystyle & = & \displaystyle \sup _{x^{\ast }\in X^{\ast }}\langle x^{\ast },\cdot \rangle +h(x^{\ast }).\nonumber\end{eqnarray}$$
We conclude that
$f=(-h)^{\ast }$
with
$\text{dom}(-h)=D$
(respectively
$\text{dom}(-h)\subset D$
), hence
$f\in \widehat{\unicode[STIX]{x1D6F4}}_{D}^{\ast }$
(respectively
$f\in \unicode[STIX]{x1D6F4}_{D}^{\ast }$
).◻
Example 7.1. Take
$D=\{a_{1}^{\ast },\ldots ,a_{n}^{\ast }\}\subset X^{\ast }$
for some
$n\geqslant 1$
. The previous proposition shows that, for every function
$f:X\rightarrow \overline{\mathbb{R}}$
,
$$\begin{eqnarray}\displaystyle f\in \unicode[STIX]{x1D6F4}_{D}^{\ast }\;\Longleftrightarrow \;f=\sup _{i\in \{1,\ldots ,n\}}\langle a_{i}^{\ast },\cdot \rangle +\unicode[STIX]{x1D6FC}_{i}\quad \text{for some }\unicode[STIX]{x1D6FC}_{1},\ldots ,\unicode[STIX]{x1D6FC}_{n}\in \overline{\mathbb{R}}. & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$
On the other hand, if
$f\in \unicode[STIX]{x1D6E4}_{0}(X)$
, the following equivalence holds true:
$$\begin{eqnarray}\text{dom}\,f^{\ast }\subset D\;\Longleftrightarrow \;\text{dom}\,f^{\ast }\subset \{a_{i}^{\ast }\}\quad \text{for some }i\in \{1,\ldots ,n\}\end{eqnarray}$$
because the set
$\text{dom}\,f^{\ast }$
is convex. Since
$f^{\ast }$
is proper, this is, in turn, equivalent to
$f^{\ast }=\unicode[STIX]{x1D6FF}_{\{a_{i}^{\ast }\}}-\unicode[STIX]{x1D6FC}_{i}$
for some
$\unicode[STIX]{x1D6FC}_{i}\in \mathbb{R}$
. Taking the conjugate, we find
$f=\langle a_{i}^{\ast },\cdot \rangle +\unicode[STIX]{x1D6FC}_{i}$
. It ensues that the set
$\{f\in \unicode[STIX]{x1D6E4}_{0}(X),\text{dom}\,f^{\ast }\subset D\}$
coincides with the set of affine continuous functions with slopes in
$D=\{a_{1}^{\ast },\ldots ,a_{n}^{\ast }\}$
. This yields an example for which the inclusion (14) is strict. By applying again Proposition 7.1, we obtain that
$$\begin{eqnarray}\displaystyle f\in \unicode[STIX]{x1D6F4}_{\text{co}(D)}^{\ast }\;\Longleftrightarrow \;f=\sup _{x^{\ast }\in \text{co}(D)}\langle x^{\ast },\cdot \rangle +\unicode[STIX]{x1D6FC}_{x^{\ast }}, & & \displaystyle\end{eqnarray}$$
with
$\unicode[STIX]{x1D6FC}_{x^{\ast }}\in \overline{\mathbb{R}}$
for every
$x^{\ast }\in \text{co}(D)$
. The comparison of (45) and (46) clearly shows that the inclusion
$\unicode[STIX]{x1D6F4}_{D}^{\ast }\subset \unicode[STIX]{x1D6F4}_{\text{co}(D)}^{\ast }$
is strict as soon as the set
$D=\{a_{1}^{\ast },\ldots ,a_{n}^{\ast }\}$
is not a singleton. This easily implies that the inclusion (15) is strict for such a set
$D$
.
The next result gives several upper bounds (in the sense of inclusion) for the set
${\mathcal{E}}^{\unicode[STIX]{x1D711}}$
, respectively when
$\unicode[STIX]{x1D711}\in \unicode[STIX]{x1D6E4}(X)$
,
$\unicode[STIX]{x1D711}\in \widehat{\unicode[STIX]{x1D6F4}}_{D}^{\ast }$
and
$\unicode[STIX]{x1D711}\in \unicode[STIX]{x1D6F4}_{D}^{\ast }$
.
Corollary 7.2. Let
$X$
be a locally convex space and let
$\unicode[STIX]{x1D711}\in \unicode[STIX]{x1D6E4}(X)$
.
-
(i) The following inclusions hold true:
(47)
$$\begin{eqnarray}\displaystyle {\mathcal{E}}^{\unicode[STIX]{x1D711}}\subset \mathop{\bigcap }_{\{\unicode[STIX]{x1D713},\unicode[STIX]{x1D711}=\unicode[STIX]{x1D713}^{\ast }\}}(\widehat{\unicode[STIX]{x1D6F4}}_{\text{dom}\,\unicode[STIX]{x1D713}}^{\ast }\cup \{-\unicode[STIX]{x1D714}_{X}\})\subset \mathop{\bigcap }_{\{\unicode[STIX]{x1D713},\unicode[STIX]{x1D711}=\unicode[STIX]{x1D713}^{\ast }\}}\unicode[STIX]{x1D6F4}_{\text{dom}\,\unicode[STIX]{x1D713}}^{\ast }. & & \displaystyle\end{eqnarray}$$
-
(ii) For every subset
$D\subset X^{\ast }$
, we have
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D711}\in \widehat{\unicode[STIX]{x1D6F4}}_{D}^{\ast } & \Longleftrightarrow & \displaystyle {\mathcal{E}}^{\unicode[STIX]{x1D711}}\subset \widehat{\unicode[STIX]{x1D6F4}}_{D}^{\ast }\cup \{-\unicode[STIX]{x1D714}_{X}\}\quad \text{if }\unicode[STIX]{x1D711}\neq -\unicode[STIX]{x1D714}_{X},\nonumber\\ \displaystyle \unicode[STIX]{x1D711}\in \unicode[STIX]{x1D6F4}_{D}^{\ast } & \Longleftrightarrow & \displaystyle {\mathcal{E}}^{\unicode[STIX]{x1D711}}\subset \unicode[STIX]{x1D6F4}_{D}^{\ast }.\nonumber\end{eqnarray}$$
-
(iii) Assume that there exist families
$(a_{i}^{\ast })_{i\in I}\subset X^{\ast }$
and
$(\unicode[STIX]{x1D6FC}_{i})_{i\in I}\subset \mathbb{R}\cup \{+\infty \}$
(respectively
$\overline{\mathbb{R}}$
) such that Then for every
$$\begin{eqnarray}\unicode[STIX]{x1D711}=\sup _{i\in I}\langle a_{i}^{\ast },\cdot \rangle +\unicode[STIX]{x1D6FC}_{i}.\end{eqnarray}$$
$g\in {\mathcal{E}}^{\unicode[STIX]{x1D711}}\setminus \{-\unicode[STIX]{x1D714}_{X}\}$
(respectively
$g\in {\mathcal{E}}^{\unicode[STIX]{x1D711}}$
), there exists
$(\unicode[STIX]{x1D6FD}_{i})_{i\in I}\subset \mathbb{R}\cup \{+\infty \}$
(respectively
$\overline{\mathbb{R}}$
) such that In particular, if the set
$$\begin{eqnarray}g=\sup _{i\in I}\langle a_{i}^{\ast },\cdot \rangle +\unicode[STIX]{x1D6FD}_{i}.\end{eqnarray}$$
$I$
is finite, every
$\unicode[STIX]{x1D711}$
-envelope is polyhedral.
Proof. (i) Let
$\unicode[STIX]{x1D713}:X^{\ast }\rightarrow \overline{\mathbb{R}}$
be such that
$\unicode[STIX]{x1D711}=\unicode[STIX]{x1D713}^{\ast }$
. Assuming that
$g\in {\mathcal{E}}^{\unicode[STIX]{x1D711}}$
, Theorem 7.1 shows that 
for some
$h\in \unicode[STIX]{x1D6E4}(X^{\ast })$
. If
$h=-\unicode[STIX]{x1D714}_{X^{\ast }}$
, we have 
and therefore
$g=-\unicode[STIX]{x1D714}_{X}$
. If
$h\neq -\unicode[STIX]{x1D714}_{X^{\ast }}$
, we see that 
hence
$g\in \widehat{\unicode[STIX]{x1D6F4}}_{\text{dom}\,\unicode[STIX]{x1D713}}^{\ast }$
. We deduce the inclusion
${\mathcal{E}}^{\unicode[STIX]{x1D711}}\subset \widehat{\unicode[STIX]{x1D6F4}}_{\text{dom}\,\unicode[STIX]{x1D713}}^{\ast }\cup \{-\unicode[STIX]{x1D714}_{X}\}$
. Since this is true for every function
$\unicode[STIX]{x1D713}:X^{\ast }\rightarrow \overline{\mathbb{R}}$
such that
$\unicode[STIX]{x1D711}=\unicode[STIX]{x1D713}^{\ast }$
, the first inclusion of (47) follows. For the second inclusion, it suffices to note that
$\widehat{\unicode[STIX]{x1D6F4}}_{\text{dom}\,\unicode[STIX]{x1D713}}^{\ast }\cup \{-\unicode[STIX]{x1D714}_{X}\}\subset \unicode[STIX]{x1D6F4}_{\text{dom}\,\unicode[STIX]{x1D713}}^{\ast }$
.
(ii) Let us fix
$D\subset X^{\ast }$
and assume that
$\unicode[STIX]{x1D711}\in \widehat{\unicode[STIX]{x1D6F4}}_{D}^{\ast }$
. Then there exists
$\unicode[STIX]{x1D713}:X^{\ast }\rightarrow \overline{\mathbb{R}}$
such that
$\unicode[STIX]{x1D711}=\unicode[STIX]{x1D713}^{\ast }$
and
$\text{dom}\,\unicode[STIX]{x1D713}=D$
. We deduce from the first inclusion of (47) that
$$\begin{eqnarray}{\mathcal{E}}^{\unicode[STIX]{x1D711}}\subset \widehat{\unicode[STIX]{x1D6F4}}_{\text{dom}\,\unicode[STIX]{x1D713}}^{\ast }\cup \{-\unicode[STIX]{x1D714}_{X}\}=\widehat{\unicode[STIX]{x1D6F4}}_{D}^{\ast }\cup \{-\unicode[STIX]{x1D714}_{X}\}.\end{eqnarray}$$
Conversely, if
${\mathcal{E}}^{\unicode[STIX]{x1D711}}\subset \widehat{\unicode[STIX]{x1D6F4}}_{D}^{\ast }\cup \{-\unicode[STIX]{x1D714}_{X}\}$
and if
$\unicode[STIX]{x1D711}\neq -\unicode[STIX]{x1D714}_{X}$
, then we obtain
$\unicode[STIX]{x1D711}\in \widehat{\unicode[STIX]{x1D6F4}}_{D}^{\ast }$
according to the inclusion
$\unicode[STIX]{x1D711}\in {\mathcal{E}}^{\unicode[STIX]{x1D711}}$
. The proof of the second equivalence is analogous and left to the reader.
(iii) Let
$(a_{i}^{\ast })_{i\in I}\subset X^{\ast }$
and
$(\unicode[STIX]{x1D6FC}_{i})_{i\in I}\subset \mathbb{R}\cup \{+\infty \}$
(respectively
$\overline{\mathbb{R}}$
) be such that
$\unicode[STIX]{x1D711}=\sup _{i\in I}\langle a_{i}^{\ast },\cdot \rangle +\unicode[STIX]{x1D6FC}_{i}.$
Let us set
$D=\{a_{i}^{\ast },i\in I\}$
. Proposition 7.1 shows that
$\unicode[STIX]{x1D711}\in \widehat{\unicode[STIX]{x1D6F4}}_{D}^{\ast }$
(respectively
$\unicode[STIX]{x1D6F4}_{D}^{\ast }$
). If
$g\in {\mathcal{E}}^{\unicode[STIX]{x1D711}}\setminus \{-\unicode[STIX]{x1D714}_{X}\}$
(respectively
$g\in {\mathcal{E}}^{\unicode[STIX]{x1D711}}$
), we deduce from (ii) that
$g\in \widehat{\unicode[STIX]{x1D6F4}}_{D}^{\ast }$
(respectively
$\unicode[STIX]{x1D6F4}_{D}^{\ast }$
). By applying Proposition 7.1 again, we derive the existence of
$(\unicode[STIX]{x1D6FD}_{i})_{i\in I}\subset \mathbb{R}\cup \{+\infty \}$
(respectively
$\overline{\mathbb{R}}$
) such that
$g=\sup _{i\in I}\langle a_{i}^{\ast },\cdot \rangle +\unicode[STIX]{x1D6FD}_{i}.$
Finally, if the set
$I$
is finite and if
$g$
is a
$\unicode[STIX]{x1D711}$
-envelope, then either
$g=\pm \unicode[STIX]{x1D714}_{X}$
or the function
$g$
is the supremum of a finite collection of continuous affine functions. We then conclude that
$g$
is polyhedral.◻
By applying the second equivalence of Corollary 7.2(ii) with
$D=X^{\ast }$
, we obtain that
$\unicode[STIX]{x1D711}\in \unicode[STIX]{x1D6E4}(X)$
if and only if
${\mathcal{E}}^{\unicode[STIX]{x1D711}}\subset \unicode[STIX]{x1D6E4}(X)$
. Corollary 7.3 below shows that in this case the set
${\mathcal{E}}^{\unicode[STIX]{x1D711}}$
is strictly included in
$\unicode[STIX]{x1D6E4}(X)$
. Note that for
$\unicode[STIX]{x1D711}\in \unicode[STIX]{x1D6E4}_{0}(X)$
satisfying a suitable condition (named generating condition), the functions of the class
${\mathcal{E}}^{\unicode[STIX]{x1D711}}$
have been studied in [Reference Polovinkin25] under the terminology of
$\unicode[STIX]{x1D711}$
-strongly convex functions.
Following Theorem 7.1 and Remark 7.1, we have
$g\in {\mathcal{E}}^{\unicode[STIX]{x1D711}}$
if and only if 
for some
$h\in \unicode[STIX]{x1D6E4}(X^{\ast })$
. Let us now have a look at the class of the functions equal to 
for some
$h:X^{\ast }\rightarrow \mathbb{R}\cup \{+\infty \}$
not necessarily in
$\unicode[STIX]{x1D6E4}(X^{\ast })$
.
Proposition 7.2. Let
$X$
be a locally convex space. Assume that
$\unicode[STIX]{x1D711}\in \unicode[STIX]{x1D6E4}_{0}(X)$
and
$g\in \unicode[STIX]{x1D6E4}_{0}(X)$
.
-
(i) If
for some
$h:X^{\ast }\rightarrow \mathbb{R}\cup \{+\infty \}$
, then we have
$g^{\infty }=\unicode[STIX]{x1D711}^{\infty }$
, which is equivalent to
$\text{cl}^{w\ast }(\text{dom}\,g^{\ast })=\text{cl}^{w\ast }(\text{dom}\,\unicode[STIX]{x1D711}^{\ast })$
. -
(ii) If
$\text{dom}\,g^{\ast }=\text{dom}\,\unicode[STIX]{x1D711}^{\ast }$
, then for
$h:X^{\ast }\rightarrow \mathbb{R}\cup \{+\infty \}$
given by
for some 
for 
Proof. (i) Assume that 
for some
$h:X^{\ast }\rightarrow \mathbb{R}\cup \{+\infty \}$
. By definition of the Legendre–Fenchel transform, we obtain
Observe that the function
$h$
cannot take the value
$+\infty$
on
$\text{dom}\,\unicode[STIX]{x1D711}^{\ast }$
(otherwise we would have
$g=\unicode[STIX]{x1D714}_{X}$
). Therefore, the values
$-\unicode[STIX]{x1D711}^{\ast }(\unicode[STIX]{x1D709}^{\ast })$
and
$h(\unicode[STIX]{x1D709}^{\ast })$
are finite for every
$\unicode[STIX]{x1D709}^{\ast }\in \text{dom}\,\unicode[STIX]{x1D711}^{\ast }$
. By taking the recession function of each member of (48), we obtain
$$\begin{eqnarray}g^{\infty }=\sup _{\unicode[STIX]{x1D709}^{\ast }\in \text{dom}\,\unicode[STIX]{x1D711}^{\ast }}[\langle \unicode[STIX]{x1D709}^{\ast },\cdot \rangle +h(\unicode[STIX]{x1D709}^{\ast })-\unicode[STIX]{x1D711}^{\ast }(\unicode[STIX]{x1D709}^{\ast })]^{\infty }.\end{eqnarray}$$
The recession function of the affine map
$\langle \unicode[STIX]{x1D709}^{\ast },\cdot \rangle +h(\unicode[STIX]{x1D709}^{\ast })-\unicode[STIX]{x1D711}^{\ast }(\unicode[STIX]{x1D709}^{\ast })$
is equal to
$\langle \unicode[STIX]{x1D709}^{\ast },\cdot \rangle$
, thus implying that
$g^{\infty }=\sup _{\unicode[STIX]{x1D709}^{\ast }\in \text{dom}\,\unicode[STIX]{x1D711}^{\ast }}\langle \unicode[STIX]{x1D709}^{\ast },\cdot \rangle =\unicode[STIX]{x1D70E}_{\text{dom}\,\unicode[STIX]{x1D711}^{\ast }}$
. Recalling that
$\unicode[STIX]{x1D70E}_{\text{dom}\,\unicode[STIX]{x1D711}^{\ast }}=\unicode[STIX]{x1D711}^{\infty }$
, we deduce that
$g^{\infty }=\unicode[STIX]{x1D711}^{\infty }$
, which is in turn equivalent to the equality
$\text{cl}^{w\ast }(\text{dom}\,g^{\ast })=\text{cl}^{w\ast }(\text{dom}\,\unicode[STIX]{x1D711}^{\ast })$
, see [Reference Moreau22].
(ii) Assume that
$\text{dom}\,g^{\ast }=\text{dom}\,\unicode[STIX]{x1D711}^{\ast }$
. It is easy to check that for every
$x^{\ast }\in X^{\ast }$
,
It ensues that 
Since
$g\in \unicode[STIX]{x1D6E4}_{0}(X)$
by assumption, we have
$g=g^{\ast \ast }$
, hence 
The function 
takes its values in
$\mathbb{R}\cup \{+\infty \}$
because
$\text{dom}\,g^{\ast }=\text{dom}\,\unicode[STIX]{x1D711}^{\ast }$
.◻
Combining Theorem 7.1 and Proposition 7.2, we derive a necessary (respectively sufficient) condition for a function
$g\in \unicode[STIX]{x1D6E4}_{0}(X)$
to be a
$\unicode[STIX]{x1D711}$
-envelope.
Corollary 7.3. Let
$X$
be a locally convex space. Assume that
$\unicode[STIX]{x1D711}\in \unicode[STIX]{x1D6E4}_{0}(X)$
and
$g\in \unicode[STIX]{x1D6E4}_{0}(X)$
.
-
(i) If
$g\in {\mathcal{E}}^{\unicode[STIX]{x1D711}}$
then
$g^{\infty }=\unicode[STIX]{x1D711}^{\infty }$
. -
(ii) If
$\text{dom}\,g^{\ast }=\text{dom}\,\unicode[STIX]{x1D711}^{\ast }$
and then
$g\in {\mathcal{E}}^{\unicode[STIX]{x1D711}}$
.
then 
Proof. (i) If
$g\in {\mathcal{E}}^{\unicode[STIX]{x1D711}}$
, we deduce from Theorem 7.1 that 
for some
$h\in \unicode[STIX]{x1D6E4}(X^{\ast })$
. Since
$g\in \unicode[STIX]{x1D6E4}_{0}(X)$
by assumption, we have
$h\neq -\unicode[STIX]{x1D714}_{X^{\ast }}$
, hence the function
$h$
does not take the value
$-\infty$
. Proposition 7.2(i) then implies that
$g^{\infty }=\unicode[STIX]{x1D711}^{\infty }$
.
(ii) If
$\text{dom}\,g^{\ast }=\text{dom}\,\unicode[STIX]{x1D711}^{\ast }$
, Proposition 7.2(ii) shows that 
with 
Since
$h\in \unicode[STIX]{x1D6E4}_{0}(X^{\ast })$
by assumption, we conclude by Theorem 7.1 that
$g\in {\mathcal{E}}^{\unicode[STIX]{x1D711}}$
.◻
7.2 Klee envelopes
Let
$(X,\Vert \cdot \Vert )$
be a normed space and let
$f:X\rightarrow \overline{\mathbb{R}}$
be an extended real-valued function. For any reals
$\unicode[STIX]{x1D706}>0$
and
$p\geqslant 1$
, we define the Klee envelope of
$f$
with index
$\unicode[STIX]{x1D706}$
and power
$p$
as
$$\begin{eqnarray}\unicode[STIX]{x1D705}_{\unicode[STIX]{x1D706},p}f(x)=\sup _{y\in X}\biggl(\frac{1}{p\unicode[STIX]{x1D706}}\Vert x-y\Vert ^{p}-f(y)\biggr).\end{eqnarray}$$
In other words, we have
$\unicode[STIX]{x1D705}_{\unicode[STIX]{x1D706},p}f=f^{\unicode[STIX]{x1D711}}$
with the function
$\unicode[STIX]{x1D711}:X\rightarrow \mathbb{R}$
defined by
$\unicode[STIX]{x1D711}(x)=(1/p\unicode[STIX]{x1D706})\Vert x\Vert ^{p}$
. Applying Theorem 7.1 with
$\unicode[STIX]{x1D711}=(1/p\unicode[STIX]{x1D706})\Vert \cdot \Vert ^{p}$
and denoting by
$\Vert \cdot \Vert _{X^{\ast }}$
the dual norm on
$X^{\ast }$
we obtain the following result.
Corollary 7.4. Let
$(X,\Vert \cdot \Vert )$
be a normed space. For any
$\unicode[STIX]{x1D706}>0$
,
$p>1$
and for every function
$f:X\rightarrow \overline{\mathbb{R}}$
, we have
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D705}_{\unicode[STIX]{x1D706},p}f=\biggl(\frac{\unicode[STIX]{x1D706}^{q-1}}{q}\Vert \cdot \Vert _{X^{\ast }}^{q}-(f_{-})^{\ast }\biggr)^{\ast }, & & \displaystyle\end{eqnarray}$$
where
$q>1$
is the conjugate exponent of
$p$
. Moreover, the following assertions are equivalent:
-
(i)
$g$
is a Klee envelope with index
$\unicode[STIX]{x1D706}$
and power
$p$
; -
(ii)
$g=((\unicode[STIX]{x1D706}^{q-1}/q)\Vert \cdot \Vert _{X^{\ast }}^{q}-h)^{\ast }$
for some
$h\in \unicode[STIX]{x1D6E4}(X^{\ast })$
; -
(iii)
$g=((\unicode[STIX]{x1D706}^{q-1}/q)\Vert \cdot \Vert _{X^{\ast }}^{q}-((\unicode[STIX]{x1D706}^{q-1}/q)\Vert \cdot \Vert _{X^{\ast }}^{q}-g^{\ast })^{\ast \ast })^{\ast }$
.
These assertions are satisfied whenever the following stronger condition is fulfilled:
-
(iv)
$g\in \unicode[STIX]{x1D6E4}(X)$
and
$(\unicode[STIX]{x1D706}^{q-1}/q)\Vert \cdot \Vert _{X^{\ast }}^{q}-g^{\ast }\in \unicode[STIX]{x1D6E4}(X^{\ast })$
.
Proof. It suffices to apply Theorem 7.1 with
$\unicode[STIX]{x1D711}=(1/p\unicode[STIX]{x1D706})\Vert \cdot \Vert ^{p}$
and
$\unicode[STIX]{x1D713}=\unicode[STIX]{x1D711}^{\ast }=(\unicode[STIX]{x1D706}^{q-1}/q)\Vert \cdot \Vert _{X^{\ast }}^{q}$
. Let us now establish the implication (iv)
$\Longrightarrow$
(ii). Assume that
$g\in \unicode[STIX]{x1D6E4}(X)$
and that
$(\unicode[STIX]{x1D706}^{q-1}/q)\Vert \cdot \Vert _{X^{\ast }}^{q}-g^{\ast }\in \unicode[STIX]{x1D6E4}(X^{\ast })$
. The function
$g^{\ast }$
can be written as
$g^{\ast }=(\unicode[STIX]{x1D706}^{q-1}/q)\Vert \cdot \Vert _{X^{\ast }}^{q}-h$
for some
$h\in \unicode[STIX]{x1D6E4}(X^{\ast })$
. Since
$g\in \unicode[STIX]{x1D6E4}(X)$
by assumption, we have
$g=g^{\ast \ast }$
. Hence, we deduce that
$g=((\unicode[STIX]{x1D706}^{q-1}/q)\Vert \cdot \Vert _{X^{\ast }}^{q}-h)^{\ast }$
and assertion (ii) is proved.◻
Corollary 7.5. Let
$(X,\Vert \cdot \Vert )$
be a normed space. For every
$p>1$
and every
$C\subset X$
, the farthest distance function
$\unicode[STIX]{x1D6E5}_{C}=\sup _{y\in C}\Vert \cdot -y\Vert$
satisfies
$$\begin{eqnarray}\frac{1}{p}\unicode[STIX]{x1D6E5}_{C}^{p}=\biggl(\frac{1}{q}\Vert \cdot \Vert _{X^{\ast }}^{q}-\unicode[STIX]{x1D70E}_{-C}\biggr)^{\ast }.\end{eqnarray}$$
Proof. Observe that
$$\begin{eqnarray}\unicode[STIX]{x1D705}_{1,p}\unicode[STIX]{x1D6FF}_{C}=\sup _{y\in X}\biggl\{\frac{1}{p}\Vert \cdot -y\Vert ^{p}-\unicode[STIX]{x1D6FF}_{C}(y)\biggr\}=\sup _{y\in C}\frac{1}{p}\Vert \cdot -y\Vert ^{p}=\frac{1}{p}\unicode[STIX]{x1D6E5}_{C}^{p}.\end{eqnarray}$$
It suffices then to apply formula (49) of Corollary 7.4 with
$f=\unicode[STIX]{x1D6FF}_{C}$
and
$\unicode[STIX]{x1D706}=1$
.◻
Additional properties of the Klee envelopes can be obtained in the case when
$(X,\Vert \cdot \Vert )$
is a Hilbert space and
$p=2$
.
Theorem 7.2. Assume that
$X$
is a Hilbert space endowed with the scalar product
$\langle \cdot ,\cdot \rangle$
and the corresponding norm
$\Vert \cdot \Vert$
.
-
(a) For every
$\unicode[STIX]{x1D706}>0$
and every function
$f:~X\rightarrow ~\overline{\mathbb{R}}$
, we have (50)
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D705}_{\unicode[STIX]{x1D706},2}f & = & \displaystyle \biggl(\frac{\unicode[STIX]{x1D706}}{2}\Vert \cdot \Vert ^{2}-(f_{-})^{\ast }\biggr)^{\ast }\end{eqnarray}$$
(51)
$$\begin{eqnarray}\displaystyle & = & \displaystyle \biggl(f-\frac{1}{2\unicode[STIX]{x1D706}}\Vert \cdot \Vert ^{2}\biggr)^{\ast }\biggl(-\frac{\cdot }{\unicode[STIX]{x1D706}}\biggr)+\frac{1}{2\unicode[STIX]{x1D706}}\Vert \cdot \Vert ^{2};\end{eqnarray}$$
(52)
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D705}_{\unicode[STIX]{x1D706},2}(\unicode[STIX]{x1D705}_{\unicode[STIX]{x1D706},2}f) & = & \displaystyle \biggl(f-\frac{1}{2\unicode[STIX]{x1D706}}\Vert \cdot \Vert ^{2}\biggr)^{\ast \ast }+\frac{1}{2\unicode[STIX]{x1D706}}\Vert \cdot \Vert ^{2}.\end{eqnarray}$$
-
(b) For
$\unicode[STIX]{x1D706}>0$
and
$f:X\rightarrow \overline{\mathbb{R}}$
the following assertions are equivalent:-
(i)
$f$
is a Klee envelope with index
$\unicode[STIX]{x1D706}$
and power
$2$
; -
(ii)
$f=((\unicode[STIX]{x1D706}/2)\Vert \cdot \Vert ^{2}-h)^{\ast }$
for some
$h\in \unicode[STIX]{x1D6E4}(X)$
; -
(iii)
$f=((\unicode[STIX]{x1D706}/2)\Vert \cdot \Vert ^{2}-((\unicode[STIX]{x1D706}/2)\Vert \cdot \Vert ^{2}-f^{\ast })^{\ast \ast })^{\ast }$
; -
(iv)
$f-(1/2\unicode[STIX]{x1D706})\Vert \cdot \Vert ^{2}\in \unicode[STIX]{x1D6E4}(X)$
; -
(v)
$f\in \unicode[STIX]{x1D6E4}(X)$
and
$(\unicode[STIX]{x1D706}/2)\Vert \cdot \Vert ^{2}-f^{\ast }\in \unicode[STIX]{x1D6E4}(X)$
.
-
Proof. (a) For the equality (50), it suffices to apply Corollary 7.4 with
$p=2$
. For the equality (51), observe that for every
$x\in X$
,
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D705}_{\unicode[STIX]{x1D706},2}f(x) & = & \displaystyle \sup _{y\in X}\biggl\{\frac{1}{2\unicode[STIX]{x1D706}}\Vert x-y\Vert ^{2}-f(y)\biggr\}\nonumber\\ \displaystyle & = & \displaystyle \sup _{y\in X}\biggl\{\frac{1}{2\unicode[STIX]{x1D706}}\Vert x\Vert ^{2}+\frac{1}{2\unicode[STIX]{x1D706}}\Vert y\Vert ^{2}-\frac{1}{\unicode[STIX]{x1D706}}\langle x,y\rangle -f(y)\biggr\}\nonumber\\ \displaystyle & = & \displaystyle \biggl(f-\frac{1}{2\unicode[STIX]{x1D706}}\Vert \cdot \Vert ^{2}\biggr)^{\ast }(-x/\unicode[STIX]{x1D706})+\frac{1}{2\unicode[STIX]{x1D706}}\Vert x\Vert ^{2}.\nonumber\end{eqnarray}$$
By iterating we deduce that
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D705}_{\unicode[STIX]{x1D706},2}(\unicode[STIX]{x1D705}_{\unicode[STIX]{x1D706},2}f) & = & \displaystyle \biggl(\unicode[STIX]{x1D705}_{\unicode[STIX]{x1D706},2}f-\frac{1}{2\unicode[STIX]{x1D706}}\Vert \cdot \Vert ^{2}\biggr)^{\ast }\biggl(-\frac{\cdot }{\unicode[STIX]{x1D706}}\biggr)+\frac{1}{2\unicode[STIX]{x1D706}}\Vert \cdot \Vert ^{2}\nonumber\\ \displaystyle & = & \displaystyle \biggl[\biggl(f-\frac{1}{2\unicode[STIX]{x1D706}}\Vert \cdot \Vert ^{2}\biggr)^{\ast }\biggl(-\frac{\cdot }{\unicode[STIX]{x1D706}}\biggr)\biggr]^{\ast }\biggl(-\frac{\cdot }{\unicode[STIX]{x1D706}}\biggr)+\frac{1}{2\unicode[STIX]{x1D706}}\Vert \cdot \Vert ^{2}\nonumber\\ \displaystyle & = & \displaystyle \biggl(f-\frac{1}{2\unicode[STIX]{x1D706}}\Vert \cdot \Vert ^{2}\biggr)^{\ast \ast }+\frac{1}{2\unicode[STIX]{x1D706}}\Vert \cdot \Vert ^{2},\nonumber\end{eqnarray}$$
which proves the equality (52).
(b) We now show that assertions (i) to (v) are equivalent. The equivalences (i)
$\Longleftrightarrow$
(ii)
$\Longleftrightarrow$
(iii) are consequences of Corollary 7.4 applied with
$p=2$
. Let us show the equivalence (i)
$\Longleftrightarrow$
(iv). Observe that
$f$
is a Klee envelope with index
$\unicode[STIX]{x1D706}$
and power
$2$
if and only if
$f\in {\mathcal{E}}^{\unicode[STIX]{x1D711}}$
with
$\unicode[STIX]{x1D711}=(1/2\unicode[STIX]{x1D706})\Vert \cdot \Vert ^{2}$
. From the equivalence (7)
$\Leftrightarrow$
(8) and the fact that
$\unicode[STIX]{x1D711}_{-}=\unicode[STIX]{x1D711}$
, this is, in turn, equivalent to
$f=(f^{\unicode[STIX]{x1D711}})^{\unicode[STIX]{x1D711}}$
. Since
$(f^{\unicode[STIX]{x1D711}})^{\unicode[STIX]{x1D711}}=\unicode[STIX]{x1D705}_{\unicode[STIX]{x1D706},2}(\unicode[STIX]{x1D705}_{\unicode[STIX]{x1D706},2}f)$
and using the equality (52), we infer that
$$\begin{eqnarray}\displaystyle & \displaystyle f\text{ is a Klee envelope with index }\unicode[STIX]{x1D706}\text{ and power }2 & \displaystyle \nonumber\\ \displaystyle & \displaystyle \Updownarrow & \displaystyle \nonumber\\ \displaystyle & \displaystyle f-\frac{1}{2\unicode[STIX]{x1D706}}\Vert \cdot \Vert ^{2}=\biggl(f-\frac{1}{2\unicode[STIX]{x1D706}}\Vert \cdot \Vert ^{2}\biggr)^{\ast \ast } & \displaystyle \nonumber\\ \displaystyle & \displaystyle \Updownarrow & \displaystyle \nonumber\\ \displaystyle & \displaystyle f-\frac{1}{2\unicode[STIX]{x1D706}}\Vert \cdot \Vert ^{2}\in \unicode[STIX]{x1D6E4}(X). & \displaystyle \nonumber\end{eqnarray}$$
Hence the equivalence (i)
$\Longleftrightarrow$
(iv) is proved. Let us now show that (iv)
$\Longrightarrow$
(v). If
$f-(1/2\unicode[STIX]{x1D706})\Vert \cdot \Vert ^{2}=\pm \unicode[STIX]{x1D714}_{X}$
, then assertion (v) is trivially satisfied. Hence we can assume that
$f=(1/2\unicode[STIX]{x1D706})\Vert \cdot \Vert ^{2}+h$
with
$h\in \unicode[STIX]{x1D6E4}_{0}(X)$
. This clearly implies that
$f\in \unicode[STIX]{x1D6E4}_{0}(X)$
. Taking the conjugate, we obtain that
$f^{\ast }=(\unicode[STIX]{x1D706}/2)\Vert \cdot \Vert ^{2}{\bigtriangledown}h^{\ast }$
since the classical qualification condition is satisfied. It ensues that for every
$x\in X$
,
$$\begin{eqnarray}\displaystyle f^{\ast }(x) & = & \displaystyle \inf _{y\in X}\biggl\{\frac{\unicode[STIX]{x1D706}}{2}\Vert x-y\Vert ^{2}+h^{\ast }(y)\biggr\}\nonumber\\ \displaystyle & = & \displaystyle \frac{\unicode[STIX]{x1D706}}{2}\Vert x\Vert ^{2}+\inf _{y\in X}\biggl\{-\unicode[STIX]{x1D706}\langle x,y\rangle +\frac{\unicode[STIX]{x1D706}}{2}\Vert y\Vert ^{2}+h^{\ast }(y)\biggr\}.\nonumber\end{eqnarray}$$
Therefore,
$$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x1D706}}{2}\Vert x\Vert ^{2}-f^{\ast }(x) & = & \displaystyle \sup _{y\in X}\biggl\{\unicode[STIX]{x1D706}\langle x,y\rangle -\frac{\unicode[STIX]{x1D706}}{2}\Vert y\Vert ^{2}-h^{\ast }(y)\biggr\}\nonumber\\ \displaystyle & = & \displaystyle \biggl(h^{\ast }+\frac{\unicode[STIX]{x1D706}}{2}\Vert \cdot \Vert ^{2}\biggr)^{\ast }(\unicode[STIX]{x1D706}x).\nonumber\end{eqnarray}$$
This clearly implies that
$(\unicode[STIX]{x1D706}/2)\Vert \cdot \Vert ^{2}-f^{\ast }\in \unicode[STIX]{x1D6E4}_{0}(X)$
and (v) is proved. Let us finally observe that the implication (v)
$\Longrightarrow$
(ii) has been established in Corollary 7.4. As a conclusion, we have shown the equivalences (i)
$\Longleftrightarrow$
(ii)
$\Longleftrightarrow$
(iii)
$\Longleftrightarrow$
(iv) along with the implications (iv)
$\Longrightarrow$
(v)
$\Longrightarrow$
(ii), which clearly establishes that all assertions (i) to (v) are equivalent.◻
The equalities (51) and (52) have been previously established by Wang [Reference Wang39] respectively in Proposition 4.5 and at the end of the proof of Proposition 4.13. As noted in [Reference Wang39, Proposition 4.13] those equalities directly yield, for
$f$
proper and lower semicontinuous, that
$\unicode[STIX]{x1D705}_{\unicode[STIX]{x1D706},2}(\unicode[STIX]{x1D705}_{\unicode[STIX]{x1D706},2}f)=f$
if and only if
$f-(1/2\unicode[STIX]{x1D706})\Vert \cdot \Vert ^{2}$
is convex.
Taking
$f$
as the indicator function of a set
$C$
gives the following corollary.
Corollary 7.6. Assume that
$X$
is a Hilbert space. For every
$C\subset X$
, the farthest distance function
$\unicode[STIX]{x1D6E5}_{C}$
satisfies
$$\begin{eqnarray}\frac{1}{2}\unicode[STIX]{x1D6E5}_{C}^{2}=(\frac{1}{2}\Vert \cdot \Vert ^{2}-\unicode[STIX]{x1D70E}_{-C})^{\ast }=(\unicode[STIX]{x1D6FF}_{-C}-\frac{1}{2}\Vert \cdot \Vert ^{2})^{\ast }+\frac{1}{2}\Vert \cdot \Vert ^{2}.\end{eqnarray}$$
7.3 Case of a positively homogeneous function
$\unicode[STIX]{x1D711}$
In this subsection, we assume that
$X$
is a locally convex space and that the function
$\unicode[STIX]{x1D711}\in \unicode[STIX]{x1D6E4}_{0}(X)$
is positively homogeneous, i.e.
$\unicode[STIX]{x1D711}=\unicode[STIX]{x1D70E}_{D}$
for a non-empty set
$D\subset X^{\ast }$
. By applying Theorem 7.1 with
$\unicode[STIX]{x1D713}=\unicode[STIX]{x1D6FF}_{D}$
, we immediately obtain the following result.
Corollary 7.7. Let
$X$
be a locally convex space. Take
$\unicode[STIX]{x1D711}=\unicode[STIX]{x1D70E}_{D}$
for a non-empty set
$D\subset X^{\ast }$
. Then we have, for every function
$f:X\rightarrow \overline{\mathbb{R}}$
,
Moreover,
Let us now particularize to the case of a normed space
$(X,\Vert \cdot \Vert )$
and take
$\unicode[STIX]{x1D711}=\Vert \cdot \Vert$
.
Corollary 7.8. Let
$(X,\Vert \cdot \Vert )$
be a normed space. For every function
$f:X\rightarrow \overline{\mathbb{R}}$
, we have
Moreover,
Proof. For the equalities 
and 
use Corollary 7.7 respectively with
$D=\mathbb{B}_{X^{\ast }}$
and
$D=\mathbb{S}_{X^{\ast }}$
. The characterizations of Klee envelopes with index
$1$
and power
$1$
follow immediately.◻
Assuming that
$f=\unicode[STIX]{x1D6FF}_{C}$
, we have
$$\begin{eqnarray}\unicode[STIX]{x1D705}_{1,1}\unicode[STIX]{x1D6FF}_{C}=\sup _{x\in X}\{\Vert \cdot -x\Vert -\unicode[STIX]{x1D6FF}_{C}(x)\}=\sup _{x\in C}\Vert \cdot -x\Vert =\unicode[STIX]{x1D6E5}_{C},\end{eqnarray}$$
where
$\unicode[STIX]{x1D6E5}_{C}$
is the farthest distance function. Taking into account the previous corollary, we then obtain
It is interesting to compare this expression with that of the signed distance
$\text{sgd}$
defined by
$\text{sgd}(\cdot ,C):=d(\cdot ,C)-d(\cdot ,X\setminus C)$
, for which it is known that
$\text{sgd}(\cdot ,C)=(\unicode[STIX]{x1D6FF}_{\mathbb{S}_{X^{\ast }}}+\unicode[STIX]{x1D70E}_{C})^{\ast },$
see [Reference Penot23].
Consider now the case of a finite set
$D=\{a_{1}^{\ast },\ldots ,a_{n}^{\ast }\}\subset X^{\ast }$
for
$n\geqslant 1$
. By applying Corollary 7.7, we obtain the following result.
Corollary 7.9. Let
$X$
be a locally convex space. Take
$\unicode[STIX]{x1D711}=\unicode[STIX]{x1D70E}_{\{a_{1}^{\ast },\ldots ,a_{n}^{\ast }\}}$
with
$a_{1}^{\ast },\ldots ,a_{n}^{\ast }\in X^{\ast }$
and
$n\geqslant 1$
. Then we have, for every function
$f:X\rightarrow \overline{\mathbb{R}}$
,
$$\begin{eqnarray}f^{\unicode[STIX]{x1D711}}=\sup _{i\in \{1,\ldots ,n\}}\langle a_{i}^{\ast },\cdot \rangle +f^{\ast }(-a_{i}^{\ast }).\end{eqnarray}$$
Moreover,
$$\begin{eqnarray}g\in {\mathcal{E}}^{\unicode[STIX]{x1D711}}\;\Longleftrightarrow \;g=\sup _{i\in \{1,\ldots ,n\}}\langle a_{i}^{\ast },\cdot \rangle +h(a_{i}^{\ast })\quad \text{for some }h\in \unicode[STIX]{x1D6E4}(X^{\ast }).\end{eqnarray}$$
8 Case
$\unicode[STIX]{x1D711}\in -\unicode[STIX]{x1D6E4}(X)$
8.1 Links between
$\unicode[STIX]{x1D711}$
-envelopes and Legendre–Fenchel conjugates
Proposition 8.1. Let
$X$
be a locally convex space and let
$\unicode[STIX]{x1D711}$
,
$g:X\rightarrow \overline{\mathbb{R}}$
be extended real-valued functions.
-
(i) If
$g\in {\mathcal{E}}^{\unicode[STIX]{x1D711}}$
, then there exists
$h\in \unicode[STIX]{x1D6E4}(X^{\ast })$
such that If, in addition,
$g\in -\unicode[STIX]{x1D6E4}(X)$
, then
-
(ii) Assume that
$X$
is normed. If
$\unicode[STIX]{x1D711}\in -\unicode[STIX]{x1D6E4}(X)$
and if there exists
$h\in \unicode[STIX]{x1D6E4}(X^{\ast })$
satisfying the equality along with the condition
$0\in \text{int}(\text{dom}\,h-\text{dom}(-\unicode[STIX]{x1D711})^{\ast })$
, then
$g\in {\mathcal{E}}^{\unicode[STIX]{x1D711}}$
.
If, in addition, 
along with the condition 
Proof. (i) Since
$g\in {\mathcal{E}}^{\unicode[STIX]{x1D711}}$
, there exists
$f:X\rightarrow \overline{\mathbb{R}}$
such that
$g=f^{\unicode[STIX]{x1D711}}$
, hence
$-g=(-\unicode[STIX]{x1D711}){\bigtriangledown}f$
by (3). Taking the conjugate of each member, we find 
Hence, the expected equality holds with
$h=f^{\ast }\in \unicode[STIX]{x1D6E4}(X^{\ast })$
. If, in addition,
$g\in -\unicode[STIX]{x1D6E4}(X)$
, we have
$-g=(-g)^{\ast \ast }$
, hence we deduce from what precedes that 
(ii) Assume that 
for some
$h\in \unicode[STIX]{x1D6E4}(X^{\ast })$
. If
$h=-\unicode[STIX]{x1D714}_{X^{\ast }}$
or if
$(-\unicode[STIX]{x1D711})^{\ast }=-\unicode[STIX]{x1D714}_{X^{\ast }}$
, then
$-g=(-\unicode[STIX]{x1D714}_{X^{\ast }})^{\ast }=\unicode[STIX]{x1D714}_{X}$
and the inclusion
$g\in {\mathcal{E}}^{\unicode[STIX]{x1D711}}$
trivially holds. Now assume that
$h\neq -\unicode[STIX]{x1D714}_{X^{\ast }}$
and
$(-\unicode[STIX]{x1D711})^{\ast }\neq -\unicode[STIX]{x1D714}_{X^{\ast }}$
. Since
$0\in \text{int}(\text{dom}\,h-\text{dom}(-\unicode[STIX]{x1D711})^{\ast })$
, the functions
$(-\unicode[STIX]{x1D711})^{\ast }$
and
$h$
are proper and according to the fact that
$X^{\ast }$
is a Banach space, we have
$$\begin{eqnarray}\displaystyle -g & = & \displaystyle (-\unicode[STIX]{x1D711})^{\ast \ast }{\bigtriangledown}h^{\ast }\nonumber\\ \displaystyle & = & \displaystyle (-\unicode[STIX]{x1D711}){\bigtriangledown}h^{\ast }\quad \text{because }\unicode[STIX]{x1D711}\in -\unicode[STIX]{x1D6E4}(X).\nonumber\end{eqnarray}$$
We conclude that
$g=\unicode[STIX]{x1D711}\triangle (-h^{\ast })=(h^{\ast })^{\unicode[STIX]{x1D711}}\in {\mathcal{E}}^{\unicode[STIX]{x1D711}}$
.◻
Corollary 8.1. Let
$X$
be a normed space and let
$\unicode[STIX]{x1D711}\in -\unicode[STIX]{x1D6E4}_{0}(X)$
be such that
$\text{dom}(-\unicode[STIX]{x1D711})^{\ast }=X^{\ast }$
. For every
$g\in -\unicode[STIX]{x1D6E4}(X)$
, the following equivalences hold true:
$$\begin{eqnarray}\displaystyle g\in {\mathcal{E}}^{\unicode[STIX]{x1D711}} & \Longleftrightarrow & \displaystyle (-g)^{\ast }-(-\unicode[STIX]{x1D711})^{\ast }\in \unicode[STIX]{x1D6E4}(X^{\ast })\nonumber\\ \displaystyle & \Longleftrightarrow & \displaystyle -g=((-\unicode[STIX]{x1D711})^{\ast }+h)^{\ast }\quad \text{for some }h\in \unicode[STIX]{x1D6E4}(X^{\ast }).\nonumber\end{eqnarray}$$
Proof. Fix
$g\in -\unicode[STIX]{x1D6E4}(X)$
. Since
$\text{dom}(-\unicode[STIX]{x1D711})^{\ast }=X^{\ast }$
and
$-\unicode[STIX]{x1D711}\in \unicode[STIX]{x1D6E4}_{0}(X)$
, the function
$(-\unicode[STIX]{x1D711})^{\ast }$
is finite-valued on
$X^{\ast }$
, so the implication
$$\begin{eqnarray}g\in {\mathcal{E}}^{\unicode[STIX]{x1D711}}\Longrightarrow h:=(-g)^{\ast }-(-\unicode[STIX]{x1D711})^{\ast }\in \unicode[STIX]{x1D6E4}(X^{\ast })\end{eqnarray}$$
follows from Proposition 8.1(i). Recalling that
$g\in -\unicode[STIX]{x1D6E4}(X)$
, the right-hand inclusion implies in turn that
$-g=((-\unicode[STIX]{x1D711})^{\ast }+h)^{\ast }$
.
Now assume that
$-g=((-\unicode[STIX]{x1D711})^{\ast }+h)^{\ast }$
for some
$h\in \unicode[STIX]{x1D6E4}(X^{\ast })$
. If
$\text{dom}\,h\neq \emptyset$
, the qualification assumption
$0\in \text{int}(\text{dom}\,h-\text{dom}(-\unicode[STIX]{x1D711})^{\ast })$
is automatically satisfied. We then deduce from Proposition 8.1(ii) that
$g\in {\mathcal{E}}^{\unicode[STIX]{x1D711}}$
. On the other hand, if
$\text{dom}\,h=\emptyset$
, then we have
$h=\unicode[STIX]{x1D714}_{X^{\ast }}$
and hence
$-g=(\unicode[STIX]{x1D714}_{X^{\ast }})^{\ast }=-\unicode[STIX]{x1D714}_{X}$
. Then the inclusion
$g\in {\mathcal{E}}^{\unicode[STIX]{x1D711}}$
trivially holds.◻
8.2 Moreau envelopes
Let
$(X,\Vert \cdot \Vert )$
be a normed space and let
$f:X\rightarrow \overline{\mathbb{R}}$
be an extended real-valued function. For
$\unicode[STIX]{x1D706}>0$
and
$p\geqslant 1$
, we define the Moreau envelope of
$f$
with index
$\unicode[STIX]{x1D706}$
and power
$p$
as
$$\begin{eqnarray}e_{\unicode[STIX]{x1D706},p}f=\inf _{y\in X}\biggl(\frac{1}{p\unicode[STIX]{x1D706}}\Vert \cdot -y\Vert ^{p}+f(y)\biggr)=\frac{1}{p\unicode[STIX]{x1D706}}\Vert \cdot \Vert ^{p}{\bigtriangledown}f.\end{eqnarray}$$
Observe that
$-e_{\unicode[STIX]{x1D706},p}f=(-(1/p\unicode[STIX]{x1D706})\Vert \cdot \Vert ^{p})\triangle (-f)=f^{\unicode[STIX]{x1D711}}$
, with the function
$\unicode[STIX]{x1D711}:X\rightarrow \mathbb{R}$
defined by
$\unicode[STIX]{x1D711}=-(1/p\unicode[STIX]{x1D706})\Vert \cdot \Vert ^{p}$
. It ensues that
$g$
is a Moreau envelope with index
$\unicode[STIX]{x1D706}$
and power
$p$
if and only if
$-g\in {\mathcal{E}}^{\unicode[STIX]{x1D711}}$
, for
$\unicode[STIX]{x1D711}=-(1/p\unicode[STIX]{x1D706})\Vert \cdot \Vert ^{p}$
. By applying the results of the previous subsection with
$\unicode[STIX]{x1D711}=-(1/p\unicode[STIX]{x1D706})\Vert \cdot \Vert ^{p}$
, we obtain the following statement.
Corollary 8.2. Assume that
$(X,\Vert \cdot \Vert )$
is a normed space. Let
$\unicode[STIX]{x1D706}>0$
,
$p>1$
and let
$q$
be the conjugate exponent of
$p$
.
-
(i) If
$g$
is a Moreau envelope with index
$\unicode[STIX]{x1D706}$
and power
$p$
, then the function
$g^{\ast }-(\unicode[STIX]{x1D706}^{q-1}/q)\Vert \cdot \Vert _{X^{\ast }}^{q}~\in ~\unicode[STIX]{x1D6E4}(X^{\ast })$
. -
(ii) If moreover
$g\in \unicode[STIX]{x1D6E4}(X)$
, the following equivalences hold true:
$$\begin{eqnarray}\displaystyle & \displaystyle g\text{ is a Moreau envelope with index }\unicode[STIX]{x1D706}\text{ and power }p & \displaystyle \nonumber\\ \displaystyle & \displaystyle \Updownarrow & \displaystyle \nonumber\\ \displaystyle & \displaystyle g^{\ast }-\frac{\unicode[STIX]{x1D706}^{q-1}}{q}\Vert \cdot \Vert _{X^{\ast }}^{q}\in \unicode[STIX]{x1D6E4}(X^{\ast }) & \displaystyle \nonumber\\ \displaystyle & \displaystyle \Updownarrow & \displaystyle \nonumber\\ \displaystyle & \displaystyle g=\biggl(\frac{\unicode[STIX]{x1D706}^{q-1}}{q}\Vert \cdot \Vert _{X^{\ast }}^{q}+h\biggr)^{\ast }\quad \text{for some }h\in \unicode[STIX]{x1D6E4}(X^{\ast }). & \displaystyle \nonumber\end{eqnarray}$$
Proof. (i) It suffices to apply Proposition 8.1(i) with
$\unicode[STIX]{x1D711}=-(1/p\unicode[STIX]{x1D706})\Vert \cdot \Vert ^{p}$
and to recall that
$((1/p\unicode[STIX]{x1D706})\Vert \cdot \Vert ^{p})^{\ast }=(\unicode[STIX]{x1D706}^{q-1}/q)\Vert \cdot \Vert _{X^{\ast }}^{q}$
.
(ii) The equivalences follow from Corollary 8.1 applied with
$$\begin{eqnarray}\unicode[STIX]{x1D711}=-\frac{1}{p\unicode[STIX]{x1D706}}\Vert \cdot \Vert ^{p}.\end{eqnarray}$$
When
$X$
is a Hilbert space, we obtain a more precise characterization of Moreau envelopes with power
$2$
, as shown by the following proposition.
Proposition 8.2. Assume that
$X$
is a Hilbert space endowed with the scalar product
$\langle \cdot ,\cdot \rangle$
and the corresponding norm
$\Vert \cdot \Vert$
.
-
(a) For every
$\unicode[STIX]{x1D706}>0$
and every function
$f:~X\rightarrow ~\overline{\mathbb{R}}$
, we have (53)The
$$\begin{eqnarray}\displaystyle e_{\unicode[STIX]{x1D706},2}f=-\biggl(f+\frac{1}{2\unicode[STIX]{x1D706}}\Vert \cdot \Vert ^{2}\biggr)^{\ast }\biggl(\frac{\cdot }{\unicode[STIX]{x1D706}}\biggr)+\frac{1}{2\unicode[STIX]{x1D706}}\Vert \cdot \Vert ^{2}. & & \displaystyle\end{eqnarray}$$
$\unicode[STIX]{x1D706}$
-proximal hull of
$f$
defined by
$h_{\unicode[STIX]{x1D706}}f=-e_{\unicode[STIX]{x1D706},2}(-e_{\unicode[STIX]{x1D706},2}f)$
is given by (54)
$$\begin{eqnarray}\displaystyle -e_{\unicode[STIX]{x1D706},2}(-e_{\unicode[STIX]{x1D706},2}f)=\biggl(f+\frac{1}{2\unicode[STIX]{x1D706}}\Vert \cdot \Vert ^{2}\biggr)^{\ast \ast }-\frac{1}{2\unicode[STIX]{x1D706}}\Vert \cdot \Vert ^{2}. & & \displaystyle\end{eqnarray}$$
-
(b) A function
$f:X\rightarrow \overline{\mathbb{R}}$
is a Moreau envelope with index
$\unicode[STIX]{x1D706}$
and power
$2$
if and only if
$f-(1/2\unicode[STIX]{x1D706})\Vert \cdot \Vert ^{2}\in -\unicode[STIX]{x1D6E4}(X)$
.
Proof. (a) For every
$x\in X$
, we have
$$\begin{eqnarray}\displaystyle e_{\unicode[STIX]{x1D706},2}f(x) & = & \displaystyle \inf _{y\in X}\biggl\{\frac{1}{2\unicode[STIX]{x1D706}}\Vert x-y\Vert ^{2}+f(y)\biggr\}\nonumber\\ \displaystyle & = & \displaystyle \inf _{y\in X}\biggl\{\frac{1}{2\unicode[STIX]{x1D706}}\Vert x\Vert ^{2}+\frac{1}{2\unicode[STIX]{x1D706}}\Vert y\Vert ^{2}-\frac{1}{\unicode[STIX]{x1D706}}\langle x,y\rangle +f(y)\biggr\}\nonumber\\ \displaystyle & = & \displaystyle -\biggl(f+\frac{1}{2\unicode[STIX]{x1D706}}\Vert \cdot \Vert ^{2}\biggr)^{\ast }(x/\unicode[STIX]{x1D706})+\frac{1}{2\unicode[STIX]{x1D706}}\Vert x\Vert ^{2},\nonumber\end{eqnarray}$$
which proves the equality (53). By iterating we deduce that
$$\begin{eqnarray}\displaystyle -e_{\unicode[STIX]{x1D706},2}(-e_{\unicode[STIX]{x1D706},2}f) & = & \displaystyle \biggl(-e_{\unicode[STIX]{x1D706},2}f+\frac{1}{2\unicode[STIX]{x1D706}}\Vert \cdot \Vert ^{2}\biggr)^{\ast }\biggl(\frac{\cdot }{\unicode[STIX]{x1D706}}\biggr)-\frac{1}{2\unicode[STIX]{x1D706}}\Vert \cdot \Vert ^{2}\nonumber\\ \displaystyle & = & \displaystyle \biggl[\biggl(f+\frac{1}{2\unicode[STIX]{x1D706}}\Vert \cdot \Vert ^{2}\biggr)^{\ast }\biggl(\frac{\cdot }{\unicode[STIX]{x1D706}}\biggr)\biggr]^{\ast }\biggl(\frac{\cdot }{\unicode[STIX]{x1D706}}\biggr)-\frac{1}{2\unicode[STIX]{x1D706}}\Vert \cdot \Vert ^{2}\nonumber\\ \displaystyle & = & \displaystyle \biggl(f+\frac{1}{2\unicode[STIX]{x1D706}}\Vert \cdot \Vert ^{2}\biggr)^{\ast \ast }-\frac{1}{2\unicode[STIX]{x1D706}}\Vert \cdot \Vert ^{2},\nonumber\end{eqnarray}$$
which proves the equality (54).
(b) Observe that
$f$
is a Moreau envelope with index
$\unicode[STIX]{x1D706}$
and power
$2$
if and only if
$-f\in {\mathcal{E}}^{\unicode[STIX]{x1D711}}$
with
$\unicode[STIX]{x1D711}=-(1/2\unicode[STIX]{x1D706})\Vert \cdot \Vert ^{2}$
. From the equivalence (7)
$\Leftrightarrow$
(8) and the fact that
$\unicode[STIX]{x1D711}_{-}=\unicode[STIX]{x1D711}$
, this is, in turn, equivalent to
$-f=((-f)^{\unicode[STIX]{x1D711}})^{\unicode[STIX]{x1D711}}$
. Since
$((-f)^{\unicode[STIX]{x1D711}})^{\unicode[STIX]{x1D711}}=-e_{\unicode[STIX]{x1D706},2}(-e_{\unicode[STIX]{x1D706},2}(-f))$
and using the equality (54), we infer that
$$\begin{eqnarray}\displaystyle \hspace{3.00003pt} & \displaystyle f\text{ is a Moreau envelope with index }\unicode[STIX]{x1D706}\text{ and power 2} & \displaystyle \nonumber\\ \displaystyle & \displaystyle \Updownarrow & \displaystyle \nonumber\\ \displaystyle & \displaystyle -f+\frac{1}{2\unicode[STIX]{x1D706}}\Vert \cdot \Vert ^{2}=\biggl(-f+\frac{1}{2\unicode[STIX]{x1D706}}\Vert \cdot \Vert ^{2}\biggr)^{\ast \ast } & \displaystyle \nonumber\\ \displaystyle & \displaystyle \Updownarrow & \displaystyle \nonumber\\ \displaystyle & \displaystyle -f+\frac{1}{2\unicode[STIX]{x1D706}}\Vert \cdot \Vert ^{2}\in \unicode[STIX]{x1D6E4}(X). & \displaystyle \nonumber\\ \displaystyle & \displaystyle \Updownarrow & \displaystyle \nonumber\\ \displaystyle & \displaystyle \hspace{110.00017pt}f-\frac{1}{2\unicode[STIX]{x1D706}}\Vert \cdot \Vert ^{2}\in -\unicode[STIX]{x1D6E4}(X).\hspace{90.00014pt}\square & \displaystyle \nonumber\end{eqnarray}$$
The coupling functional
$(x,y)\mapsto -(1/2\unicode[STIX]{x1D706})\Vert x-y\Vert ^{2}$
was considered in [Reference Dolecki and Kurcyusz7, §5] in the framework of generalized conjugacy. Equalities (53)–(54) were established by Penot and Volle [Reference Penot, Volle and Cambini24, p. 206] and Martinez-Legaz [Reference Martinez-Legaz and Cambini17, pp. 182–184]. These equalities were also observed in [Reference Rockafellar and Wets30, Example 11.26(c)] and [Reference Wang39, Lemma 3.3]. The characterization (b) above has been noted in the aforementioned references, and it amounts to the previous characterization in [Reference Dolecki and Kurcyusz7, p. 288] of
$Q^{c}$
-convex functions with
$c:=1/(2\unicode[STIX]{x1D706})$
. It seems that Moreau was the first who provided this characterization (b), see the equivalence (I)
$\Leftrightarrow$
(III) in [Reference Moreau20, Proposition 9.b].
