1 Introduction
The setting of this paper is in the Euclidean n-space
$\mathbb {R}^n$
with
$n \geq 2$
. For two vectors
$x_1$
and
$x_2$
in
$\mathbb {R}^n$
, their radial sum is defined by
$$\begin{align*}x_1 \mathop{\widetilde{+}} x_2 = \begin{cases} x_1 + x_2, & \exists \left( s_1 , s_2\right) \in \mathbb{R}^2 \mathop{\backslash} \{ (0,0) \} : s_1 x_1 + s_2 x_2 =0 ,\\ 0, & \text{otherwise}. \end{cases} \end{align*}$$
Throughout this paper, we understand that every star-shaped subset of
$\mathbb {R}^n$
is star-shaped with respect to the origin. For two star-shaped subsets
$A_1$
and
$A_2$
of
$\mathbb {R}^n$
, their radial sum is defined by
$$\begin{align*}A_1 \mathop{\widetilde{+}} A_2 = \left\{ \left. x_1 \mathop{\widetilde{+}} x_2 \, \right\vert \, x_1 \in A_1 , \ x_2 \in A_2 \right\}. \end{align*}$$
A subset A of
$\mathbb {R}^n$
is centered if
$-x \in A$
for any
$x \in A$
. A subset A of
$\mathbb {R}^n$
is centrally symmetric if there exists a point
$y \in \mathbb {R}^n$
such that
$A-y = \{ x- y \, \vert \, x \in A \}$
is centered. We denote by
$r B^n$
the centered ball of radius r. Let
$S^{n-1}$
be the boundary of
$B^n = 1B^n$
. For a star-shaped subset A of
$\mathbb {R}^n$
, the radial function of A is defined by
$$\begin{align*}\rho_A (u) = \max \{ \lambda \in [ 0,+\infty ) \, \vert \, \lambda u \in A \} ,\ u \in S^{n-1}. \end{align*}$$
A body in
$\mathbb {R}^n$
is the closure of a bounded open subset of
$\mathbb {R}^n$
. A star body in
$\mathbb {R}^n$
is a star-shaped body of
$\mathbb {R}^n$
whose radial function is continuous.
It follows from the definition that the radial sum of two star-shaped subsets is star-shaped. For two star bodies
$A_1$
and
$A_2$
in
$\mathbb {R}^n$
, the relation
$\rho _{A_1 \mathop{\widetilde{+}} A_2} = \rho _{A_1} + \rho _{A_2}$
yields that
$A_1 \mathop{\widetilde{+}} A_2$
is a star body. However, the convex versions of these properties do not always hold. Namely, there exist two convex subsets/bodies containing the origin whose radial sum is not convex. Such two convex subsets are
$\{ (t , 0 , 0 , \ldots , 0 ) \, \vert \, t \in [-1,1] \}$
and
$\{ (0, t , 0, \ldots , 0 ) \, \vert \, t \in [-1, 1] \}$
. Such two convex bodies are
$\{ (t, 0 , \ldots , 0 ) + x \, \vert \, t \in [-2 \sqrt {2} , 2 \sqrt {2} ] , \ x \in B^n \}$
and
$rB^n$
for any
$r \in (0 , -1 + \sqrt {6}/2 )$
(see Example 2.2 for details).
In contrast to these examples, as a main investigation of this paper, we show the following two results:
-
(1) Let
$\gamma \in [0,+\infty )$
, and let K be a convex body containing the origin whose radial function is of class
$C^2$
. If for any
$$ \begin{align*} \left( \rho_K \left( u_1 \right) + \rho_K \left( u_2 \right) \right) \rho_K \left( \frac{u_1 + u_2}{\left\vert u_1 + u_2 \right\vert} \right) - \left\vert u_1 + u_2 \right\vert \rho_K \left( u_1 \right) \rho_K \left( u_2 \right) \geq \gamma \angle \left( u_1 ,u_2 \right)^2 \end{align*} $$
$(u_1, u_2 ) \in S^{n-1} \times S^{n-1}$
with
$u_1 + u_2 \neq 0$
, then, for any “small enough”
$f \in C^2 (S^{n-1})$
, the star body defined by
$\rho _K + f$
is convex (see Proposition 3.1 for details).
-
(2) Let A be a star body whose radial function is of class
$C^2$
. There exists a “large enough”
$R \in (0,+\infty )$
such that, for any
$r \in (0,+\infty )$
, if
$r \geq R$
, then
$A \mathop{\widetilde{+}} r B^n$
is convex (see Theorem 3.3 and its corollaries for details).
Here, in the first assertion, if
$\gamma =0$
, then the inequality for
$\rho _K$
means that K is convex (see Lemma 2.1). If
$\gamma>0$
, then the inequality means that K has a “stronger” convexity associated with
$\gamma $
, which is precisely explained right after Lemma 2.1. Any centered ball has this property. Thus, the first assertion yields that the radial sum of a “small perturbation” of any centered ball and a centered ball is convex.
The technical key point of the proofs is to approximate
$C^2$
-functions of two real variables by polynomials. As we know, Taylor’s theorem yields that any
$C^2$
-function
$\Phi $
of two real variables
$\theta _1$
and
$\theta _2$
has an approximation of the form
$$\begin{align*}\Phi \left( \theta_1 , \theta_2 \right) \approx c_0 + c_{1} \theta_1 + c_{2} \theta_2 + c_{1,1} \theta_1^2 + c_{1,2} \theta_1 \theta_2 + c_{2,2} \theta_2^2. \end{align*}$$
We improve this approximation for certain suitable
$\Phi $
and obtain an approximation of the form
$$\begin{align*}\Phi \left( \theta_1 , \theta_2 \right) \approx c_0 + c_1 \left( \theta_1 - \theta_2 \right) + c_2 \left( \theta_1 -\theta_2 \right)^2. \end{align*}$$
Using this approximation, as another main investigation of this paper, we give convex bodies with no symmetries whose intersection bodies are convex. We denote by
$V_k$
the k-dimensional Lebesgue measure. For a unit vector u, we denote by
$u^\perp $
the orthogonal complement of u. For a star body A in
$\mathbb {R}^n$
, its intersection body is denoted by
$IA$
and is the star body defined by
$$\begin{align*}\rho_{IA} (u) = V_{n-1} \left( A \cap u^\perp \right) ,\ u \in S^{n-1}. \end{align*}$$
A star body A is called an intersection body if there is a star body such that its intersection body is A. The notion of intersection body was introduced in [Reference Lutwak10] and played an important role in the solution of the Busemann–Petty problem.
In [Reference Busemann and Petty3], Busemann and Petty posed the following question:
For any centered convex bodies K and L in
$\mathbb {R}^n$
, if
$V_{n-1} (K \cap u^\perp ) \leq V_{n-1} (L \cap u^\perp )$
for any
$u \in S^{n-1}$
, then is
$V_n (K ) \leq V_n (L)$
true?
This question is now referred to as the Busemann–Petty problem and appears often in the literature about geometric tomography (see, for example, [Reference Gardner6]), in which information about a given geometric object is obtained from data concerning its sections and/or projections.
In the history of the Busemann–Petty problem, Lutwak’s theorem [Reference Lutwak10, Theorem 10.1] is important as the first step toward the full solution. It states that, for any intersection body K and any star body L, if
$V_{n-1} ( K \cap u^\perp ) \leq V_{n-1} (L \cap u^\perp )$
for any
$u \in S^{n-1}$
, then
$V_n (K ) \leq V_n (L)$
. In particular, this theorem yields that if K is an intersection body, then the answer to the Busemann–Petty problem is affirmative. It was shown in [Reference Gardner4, Theorem 3.1] with a regularity assumption and in [Reference Zhang13, Theorem 2.22] without any regularity assumption that the Busemann–Petty problem has a positive answer in
$\mathbb {R}^n$
if and only if each centered convex body in
$\mathbb {R}^n$
is an intersection body. This equivalence gives negative answers to the Busemann–Petty problem: Gardner [Reference Gardner4, Theorem 6.1] showed that if
$n \geq 5$
, then a cylinder in
$\mathbb {R}^n$
is not an intersection body. Of course, this equivalence gives also positive answers: Gardner [Reference Gardner5, Corollary 5.3] showed that every centered convex body in
$\mathbb {R}^3$
is an intersection body; Zhang [Reference Zhang14, Theorem 3] showed that every centered convex body in
$\mathbb {R}^4$
is an intersection body (see also [Reference Gardner, Koldobsky and Schlumprecht7] for an analytic approach). We refer to [Reference Gardner6, Chapter 8] for more information on the Busemann–Petty problem (see also [Reference Hansen, Herburt, Martini and Moszyńska8, Section 17] and [Reference Moszyńska12, Section 15]).
From the point of view of the Busemann–Petty problem, it is important to obtain a convex intersection body. Here, the term “a convex intersection body” means “an intersection body which is convex” and is not used in the sense of [Reference Meyer and Reisner11]. By definition, the intersection body of any star body is a centered star body. However, there exists a convex body containing the origin whose intersection body is not convex. Let us review some results on the convexity of intersection bodies. Gardner produced nonconvex intersection bodies in his textbook [Reference Gardner6, Section 8.1]. Precisely, [Reference Gardner6, Theorem 8.1.8] states that, for any convex body K, there exists a translate of K such that it contains the origin in its interior and its intersection body is not convex. Also, [Reference Gardner6, Theorem 8.1.9] remarks the following two examples:
-
(1) Let K be an equilateral triangle whose centroid is at the origin. For any
$y \in \mathbb {R}^2$
, if the interior of
$K + y$
contains the origin, then
$I(K+y)$
is not convex. -
(2) Let K be a square whose centroid is at the origin. For any
$y \in \mathbb {R}^2 \mathop{\backslash} \{ 0 \}$
, if the interior of
$K + y$
contains the origin, then
$I(K+y)$
is not convex.
In contrast to Gardner’s indication, Busemann’s theorem [Reference Busemann2] yields that the intersection body of any centered convex body is convex (see also [Reference Gardner6, Theorem 8.1.10 and Corollary 8.1.11]). As a generalization of Busemann’s theorem, it was shown in [Reference Kim, Yaskin and Zvavitch9, Theorem 3] that, for any
$p \in (0,1]$
, the intersection body of any centered p-convex body is
$1/ [(1/p-1)(n-1)+1]$
-convex. Here, we recall that a subset K of
$\mathbb {R}^n$
is p-convex if, for any
$(x_0 , x_1 ,\lambda ) \in K \times K \times [0,1]$
,
$(1-\lambda )^{1/p} x_0 + \lambda ^{1/p} x_1 \in K$
. We emphasize that central symmetry essentially works for Busemann’s theorem and its generalization. In [Reference Alfonseca and Kim1], the local convexity of intersection bodies of symmetric convex bodies of revolution was investigated. Namely, it is still open to concretely give convex bodies with no symmetries whose intersection bodies are globally convex.
We produce convex bodies with no symmetries whose intersection bodies are convex:
-
(1) Let
$\gamma \in [0,+\infty )$
, and let A be a star body whose radial function is of class
$C^2$
. If for any
$$ \begin{align*} \left( \rho_{IA} \left( u_1 \right) + \rho_{IA} \left( u_2 \right) \right) \rho_{IA} \left( \frac{u_1 + u_2}{\left\vert u_1 + u_2 \right\vert} \right) - \left\vert u_1 + u_2 \right\vert \rho_{IA} \left( u_1 \right) \rho_{IA} \left( u_2 \right) \geq \gamma \angle \left( u_1 ,u_2 \right)^2 \end{align*} $$
$(u_1, u_2 ) \in S^{n-1} \times S^{n-1}$
with
$u_1 + u_2 \neq 0$
, then, for any “small enough”
$f \in C^2 (S^{n-1})$
, the intersection body of the star body defined by
$\rho _A + f$
is convex (see Proposition 3.7 for details).
-
(2) Let A be a star body whose radial function is of class
$C^2$
. There exists a “large enough”
$\widetilde {R} \in (0,+\infty )$
such that, for any
$r \in (0,+\infty )$
, if
$r \geq \widetilde {R}$
, then the intersection body of
$A \mathop{\widetilde{+}} r B^n$
is convex (see Theorem 3.9 and its corollaries for details).
Here, in the first assertion, any centered ball has the property since the intersection body of any centered ball is a centered ball. Thus, the first assertion yields that the intersection body of a “small perturbation” of any centered ball is convex.
2 Preliminaries
2.1 Notation and terminology
Let us prepare necessary notation and terminology in addition to those stated in the Introduction.
We denote by
$\mathcal {S}^n$
the set of all star bodies (star-shaped bodies with respect to the origin whose radial functions are continuous) in
$\mathbb {R}^n$
. We denote by
$\mathcal {K}^n$
the set of all convex bodies in
$\mathbb {R}^n$
, and let
$\mathcal {K}^n_0 = \{ K \in \mathcal {K}^n \, \vert \, 0 \in K \}$
.
Let
$\kappa _k = V_k ( B^k)$
. The symbol
$\sigma _k$
denotes the k-dimensional spherical Lebesgue measure. We denote by
$C^m (M)$
the set of all
$C^m$
-functions defined on a manifold M. Let
$e_i$
be the ith unit vector of
$\mathbb {R}^n$
. Let
$SO(n)$
denote the special orthogonal group of degree n.
For a star-shaped subset A of
$\mathbb {R}^n$
, the extended radial function of A is denoted by the same symbol as the usual radial function and is defined by
$$\begin{align*}\rho_A (x ) = \begin{cases} \displaystyle \vert x \vert \rho_A \left( \frac{x}{\vert x \vert} \right), & x \in \mathbb{R}^n \mathop{\backslash} \{ 0 \} ,\\ \infty, & x =0. \end{cases} \end{align*}$$
For a star-shaped subset A of
$\mathbb {R}^n$
and a function f defined on
$S^{n-1}$
such that
$\rho _A + f \geq 0$
on
$S^{n-1}$
, let
$A_f$
be the star-shaped subset whose radial function is
$\rho _A + f$
. In particular, for a star-shaped subset A of
$\mathbb {R}^n$
and a nonnegative constant r,
$A_r = A \mathop{\widetilde{+}} r B^n$
.
For a function f defined on
$S^{n-1}$
and a nonnegative integer i, let
$f^i$
be the function
$S^{n-1} \ni u \mapsto f(u)^i$
. For two functions f and g defined on
$S^{n-1}$
, let
$f+g$
and
$fg$
be the functions
$S^{n-1} \ni u \mapsto f(u) + g(u)$
and
$S^{n-1} \ni u \mapsto f(u) g(u)$
, respectively. For a continuous function f defined on
$S^{n-1}$
, we denote by
$\mathcal {R} [f]$
the spherical Radon transform of f, that is,
$$\begin{align*}\mathcal{R} [f] (u) = \int_{S^{n-1} \cap u^\perp} f (v) \, \mathrm{d} \sigma_{n-2} (v) ,\ u \in S^{n-1}. \end{align*}$$
For two continuous functions f and g defined on
$S^{n-1}$
, put
$\left \langle f ,g \right \rangle = \mathcal {R} [fg]$
.
For two unit vectors
$u_1$
and
$u_2$
, we use the following notation:
$$\begin{align*}\angle \left( u_1 ,u_2 \right) = \arccos u_1 \cdot u_2 ,\ u_3 = \begin{cases} \displaystyle \frac{u_1 + u_2}{\left\vert u_1 + u_2 \right\vert}, & u_1 + u_2 \neq 0 ,\\ 0, & u_1 + u_2 =0. \end{cases} \end{align*}$$
For two functions f and g defined on
$S^{n-1}$
, we define
$\Delta [f,g] : S^{n-1} \times S^{n-1} \to (-\infty , \infty ]$
as:
-
(i) If
$u_1 + u_2 \neq 0$
, then
$$ \begin{align*} \Delta [f,g] \left( u_1 , u_2 \right) = \left( f \left( u_1 \right) + f \left( u_2 \right) \right) g \left( u_3 \right) + \left( g \left( u_1 \right) + g \left( u_2 \right) \right) f \left( u_3 \right) \\ - \left\vert u_1 + u_2 \right\vert \left( f \left( u_1 \right) g \left( u_2 \right) + f \left( u_2 \right) g \left( u_1 \right)\right). \end{align*} $$
-
(ii) If
$u_1 + u_2 = 0$
, then
$\Delta [f,g] (u_1 , u_2 ) = \infty $
.
For a function f defined on
$S^{n-1}$
, put
$$\begin{align*}\Gamma [ f ] \left( u_1, u_2 \right) = \frac{\Delta [f,f] \left( u_1 , u_2 \right)}{2} ,\ \left( u_1, u_2 \right) \in S^{n-1} \times S^{n-1}. \end{align*}$$
We denote by
${\| \cdot \|}_\infty $
the sup-norm for bounded functions. For a function
$\varphi $
of k real variables
$\theta _1, \ldots , \theta _k$
with bounded partial derivatives, we write
$$ \begin{align*} \varphi' = \left( \frac{\partial \varphi}{\partial \theta_1} , \ldots , \frac{\partial \varphi}{\partial \theta_k} \right) ,\ \left\Vert \varphi' \right\Vert_\infty = \max \left\{ \left. \left\Vert \frac{\partial \varphi}{\partial \theta_i} \right\Vert_\infty \, \right\vert \, i \in \{ 1, \ldots , k \} \right\}, \\ \varphi" = \left( \frac{\partial^2 \varphi}{\partial \theta_i \partial \theta_j} \right) ,\ \left\Vert \varphi" \right\Vert_\infty = \max \left\{ \left. \left\Vert \frac{\partial^2 \varphi}{\partial \theta_i \partial \theta_j} \right\Vert_\infty \, \right\vert \, i ,j \in \{ 1, \ldots , k \} \right\} . \end{align*} $$
Put
$$ \begin{align*} \phi \left( \theta_1 , \ldots , \theta_{n-1} \right) = \left( \begin{array}{c} \cos \theta_1 \\ \sin \theta_1 \cos \theta_2 \\ \vdots \\ \sin \theta_1 \cdots \sin \theta_{n-2} \cos \theta_{n-1} \\ \sin \theta_1 \cdots \sin \theta_{n-2} \sin \theta_{n-1} \end{array} \right) ,\ \left( \theta_1 , \ldots , \theta_{n-1} \right) \in \mathbb{R}^{n-1}. \end{align*} $$
We define
$\Phi _0 : C^2 ( S^{n-1} ) \times C^2 ( S^{n-1} ) \to [0,+\infty )$
as:
-
(I) On the off-diagonal set,
-
(i) if
$f(u_1) g(u_2) + f (u_2 ) g(u_1 ) < 0$
for some
$(u_1, u_2 ) \in S^{n-1} \times S^{n-1}$
, then
$$ \begin{align*} \Phi_0 ( f,g ) = 2 \left\Vert ( f \circ \phi )' \right\Vert_\infty \left\Vert ( g \circ \phi )' \right\Vert_\infty + {\| f \circ \phi \|}_\infty \left\Vert ( g \circ \phi )" \right\Vert_\infty \\ + {\Vert f \circ \phi \|}_\infty {\Vert g \circ \phi \|}_\infty; \end{align*} $$
-
(ii) if
$f(u_1) g(u_2) + f (u_2 ) g(u_1 ) \geq 0$
for every
$(u_1, u_2 ) \in S^{n-1} \times S^{n-1}$
, then
$$\begin{align*}\Phi_0 ( f,g ) = 2 \left\Vert ( f \circ \phi )' \right\Vert_\infty \left\Vert ( g \circ \phi )' \right\Vert_\infty + {\| f \circ \phi \|}_\infty \left\Vert ( g \circ \phi )" \right\Vert_\infty . \end{align*}$$
-
-
(II) On the diagonal set,
-
(i) if
$\Gamma [f] (u_1, u_2 ) <0$
for some
$(u_1, u_2 ) \in S^{n-1} \times S^{n-1}$
, and if
$f(u_1 ) f(u_2 ) <0$
for some
$(u_1, u_2) \in S^{n-1} \times S^{n-1}$
, then
$$\begin{align*}\Phi_0 (f,f) = 2 \left\Vert ( f \circ \phi )' \right\Vert_\infty^2 + {\| f \circ \phi \|}_\infty \left\Vert ( f \circ \phi )" \right\Vert_\infty + {\| f \circ \phi \| }_\infty^2 ; \end{align*}$$
-
(ii) if
$\Gamma [f] (u_1, u_2 ) <0$
for some
$(u_1, u_2 ) \in S^{n-1} \times S^{n-1}$
, and if
$f(u_1 ) f(u_2 ) \geq 0$
for every
$(u_1, u_2) \in S^{n-1} \times S^{n-1}$
, then
$$\begin{align*}\Phi_0 (f,f) = 2 \left\Vert ( f \circ \phi )' \right\Vert_\infty^2 + {\| f \circ \phi \|}_\infty \left\Vert ( f \circ \phi )" \right\Vert_\infty ; \end{align*}$$
-
(iii) if
$\Gamma [f] (u_1, u_2 ) \geq 0$
for every
$(u_1, u_2) \in S^{n-1} \times S^{n-1}$
, then
$\Phi _0 (f,f) =0$
.
-
We define
$\Phi _1 : C^2 ( S^{n-1} ) \to [0, + \infty )$
as:
-
(i) If
$f (u ) < 0$
for some
$u \in S^{n-1}$
, then
$\Phi_1 (f) = 2 {\| f \circ \phi \|}_\infty + {\| ( f \circ \phi )" \|}_\infty $
. -
(ii) If
$f ( u ) \geq 0$
for every
$u \in S^{n-1}$
, then
$\Phi_1 ( f ) = {\| ( f \circ \phi )" \|}_\infty $
.
2.2 Examples of nonconvex radial sums of convex bodies
The following lemma is useful in investigating the convexity of a star body.
Lemma 2.1 [Reference Gardner6, Lemma 5.1.4]
Let
$A \in \mathcal {S}^n$
. The following conditions are equivalent:
-
(i) A is convex.
-
(ii) For any
$( u_1, u_2 ) \in S^{n-1} \times S^{n-1}$
, we have
$\Gamma [ \rho _A ] ( u_1 , u_2 ) \geq 0$
.
For the condition (ii), we remark that if
$u_1 + u_2 =0$
, then
$\Gamma [ \rho _A ] (u_1, u_2 ) = \infty $
. Thus, the condition (ii) essentially works for the case where
$u_1+u_2 \neq 0$
. In order to understand the philosophy of Lemma 2.1, assume
$u_1+ u_2 \neq 0$
and
$\rho _A (u_1 ) \rho _A (u_2) \rho _A (u_3 ) \neq 0$
. Let us compute the unique point of the intersection
$$\begin{align*}\left\{ \left. s \rho_A \left( u_3 \right) u_3 \, \right\vert \, s \in [ 0 ,+\infty ) \right\} \cap \left\{ \left. ( 1- t ) \rho_A \left( u_1 \right) u_1 + t \rho_A \left( u_2 \right) u_2 \, \right\vert \, t \in [0,1] \right\}. \end{align*}$$
There exists a pair
$(s,t) \in [ 0,+\infty ) \times [ 0,1]$
such that
$$\begin{align*}s \rho_A \left( u_3 \right) u_3 = (1-t ) \rho_A \left( u_1 \right) u_1 + t \rho_A \left( u_2 \right) u_2 , \end{align*}$$
and we get
$$\begin{align*}s = \frac{\rho_A \left( u_1 \right) \rho_A \left( u_2 \right)}{\rho_A \left( u_1 \right) + \rho_A \left( u_2 \right)} \frac{\left\vert u_1 + u_2 \right\vert}{\rho_A \left( u_3 \right)}. \end{align*}$$
Since
$\rho _A (u_1) u_1$
,
$\rho _A (u_2) u_2$
, and
$\rho _A (u_3 ) u_3$
are on the boundary of A, A is convex if and only if
$s \leq 1$
which is equivalent to
$\Gamma [\rho _A ] (u_1, u_2 ) \geq 0$
. This finishes the proof of Lemma 2.1.
From this observation, if there is a positive function
$\varepsilon $
with
$\varepsilon ( 0^+ ) =0$
such that, for any
$( u_1 , u_2 ) \in S^{n-1} \times S^{n-1}$
,
$$\begin{align*}\frac{\rho_A \left( u_1 \right) \rho_A \left( u_2 \right)}{\rho_A \left( u_1 \right) + \rho_A \left( u_2 \right)} \frac{\left\vert u_1 + u_2 \right\vert}{\rho_A \left( u_3 \right)} \leq 1 - \varepsilon \left( \angle \left( u_1 , u_2 \right) \right) \end{align*}$$
holds, then A has a “stronger” convexity associated with
$\varepsilon $
. This observation will be used in Propositions 3.1 and 3.7 with
$\varepsilon (\theta ) = \theta ^2$
(up to constant multiple).
Using Lemma 2.1, let us concretely give a pair
$(K ,r ) \in \mathcal {K}^n_0 \times (0,+\infty )$
such that
$K \mathop{\widetilde{+}} rB^n$
is not convex.
Example 2.2 Let
$\ell \in (0 , + \infty )$
. Put
$$ \begin{align*}K &= \left\{ te_1 + x \, \left\vert \, t \in [-\ell , \ell] ,\ x \in B^n \right. \right\} ,\ \alpha (\ell) = \arctan \frac{1}{\ell} ,\\r_{\ast} (\ell) &= - \frac{2 \sin^3 \left( \frac{\alpha (\ell) }{2} + \frac{\pi}{4} \right) + \sin^2 \left( \frac{\alpha (\ell ) }{2} + \frac{\pi}{4} \right) - \sin \left( \frac{\alpha (\ell) }{2} + \frac{\pi}{4} \right) - 1}{\sin \alpha (\ell) \sin \left( \frac{\alpha (\ell ) }{2} + \frac{\pi}{4} \right)} ,\\s_{\ast} &= \frac{\left( 44+ 3 \sqrt{177} \right)^{1/3} + \left( 44 - 3 \sqrt{177} \right)^{1/3} -1}{6} ,\ \ell_{\ast} = \frac{2 s_{\ast} \sqrt{1-s_{\ast}^{2}}}{2s_{\ast}^{2}-1} . \end{align*} $$
-
(1)
$s_{\ast }$
is the unique real root of
$2s^{3}+s^{2}-s-1 =0$
, and
$0.829 < s_{\ast } < 0.830$
. -
(2)
$\ell _{\ast }$
is the unique root of
$\sin ( \alpha (\ell ) /2 + \pi /4 ) = s_{\ast }$
, and
$2.450 < \ell _{\ast } <2.476$
. -
(3) If
$\ell> \ell _{\ast }$
, then
$r_{\ast } (\ell )>0$
. -
(4) If
$\ell> \ell _{\ast }$
, then, for any
$r \in (0, r_{\ast } (\ell ))$
,
$K \mathop{\widetilde{+}} rB^n$
is not convex.
Proof (4) Let us check that the condition (ii) in Lemma 2.1 does not hold. Put
$$\begin{align*}u_1 = \left( \begin{array}{c} \cos \alpha (\ell) \\ \sin \alpha (\ell) \\ 0 \\ \vdots \\ 0 \end{array} \right) ,\ u_2 =e_2. \end{align*}$$
Then, we have
$$ \begin{align*} \left\vert u_1 + u_2 \right\vert &= 2 \cos \left( \frac{\alpha (\ell)}{2} - \frac{\pi}{4} \right) ,\ u_3 =\left( \begin{array}{c} \cos \left( \frac{\alpha (\ell)}{2} + \frac{\pi}{4} \right) \\ \sin \left( \frac{\alpha (\ell)}{2} + \frac{\pi}{4} \right) \\ 0 \\ \vdots \\ 0 \end{array} \right) ,\\\rho_K \left( u_1 \right) &= \sqrt{\ell^2+1} ,\ \rho_K \left( u_2 \right) = 1 , \ \rho_K \left( u_3 \right) = \frac{1}{\sin \left( \frac{\alpha (\ell)}{2} + \frac{\pi}{4} \right)} ,\\r_{\ast} (\ell) &= - \frac{2 \rho_K \left( u_3 \right) + \left( 1 - \left\vert u_1 + u_2 \right\vert \right) \left( \rho_K \left( u_1 \right) + \rho_K \left( u_2 \right) \right)}{2- \left\vert u_1 + u_2 \right\vert }. \end{align*} $$
Noting
$\Gamma [ \rho _K ] ( u_1, u_2 ) =0$
, we obtain
$$\begin{align*}\Gamma \left[ \rho_K + r \right] \left( u_1, u_2 \right) = \left( 2- \left\vert u_1 + u_2 \right\vert \right) r \left( r - r_{\ast} (\ell) \right), \end{align*}$$
which is negative for any
$r \in ( 0 , r_{\ast } (\ell ) )$
.
Example 2.3 Let
$A \in \mathcal {S}^2$
. By definition, we have
$$\begin{align*}I A = 2 \left( \begin{array}{cc} 0 &-1 \\ 1 & 0 \end{array} \right) A. \end{align*}$$
Combining this and Example 2.2 with
$n=2$
, we have a star body of the form
$K \mathop{\widetilde{+}} rB^2$
whose intersection body is not convex.
2.3 Technical lemmas and remarks
Lemma 2.4 Let
$I \subset \mathbb {R}$
be an open interval, and let
$\varphi $
and
$\psi \in C^2 (I)$
. For any
$( \theta _1 , \theta _2 ) \in I \times I$
, we have
$$ \begin{align*}&\left\vert \left( \varphi \left( \theta_1 \right) + \varphi \left( \theta_2 \right) \right) \psi \left( \frac{\theta_1 + \theta_2}{2} \right) - \varphi \left( \theta_1 \right) \psi \left( \theta_2 \right) - \varphi \left( \theta_2 \right) \psi \left( \theta_1 \right) \right\vert \\ &\quad \leq \left( 2 \left\Vert \varphi' \right\Vert_\infty \left\Vert \psi' \right\Vert_\infty + \left\Vert \varphi \right\Vert_\infty \left\Vert \psi" \right\Vert_\infty \right) \frac{\left( \theta_1 -\theta_2 \right)^2}{4} . \end{align*} $$
Proof By the fundamental theorem of calculus, we have
$$ \begin{align*} &\left( \varphi \left( \theta_1 \right) + \varphi \left( \theta_2 \right) \right) \psi \left( \frac{\theta_1 + \theta_2}{2} \right) - \varphi \left( \theta_1 \right) \psi \left( \theta_2 \right) -\varphi \left( \theta_2 \right) \psi \left( \theta_1 \right) \\&= \varphi \left( \theta_1 \right) \left( \psi \left( \frac{\theta_1 + \theta_2}{2} \right) - \psi \left( \theta_2 \right) \right) + \varphi \left( \theta_2 \right) \left( \psi \left( \frac{\theta_1 + \theta_2}{2} \right) - \psi \left( \theta_1 \right) \right) \\&= \frac{\theta_1 - \theta_2}{2} \int_0^1 \left( \varphi \left( \theta_1 \right) \psi' \left( s \frac{\theta_1 +\theta_2}{2} + (1-s) \theta_2 \right) \right. \\&\quad \left. - \varphi \left( \theta_2 \right) \psi' \left( s \frac{\theta_1 +\theta_2}{2} + (1-s) \theta_1 \right) \right) \mathrm{d} s . \end{align*} $$
Using the fundamental theorem of calculus again, the integrand is
$$ \begin{align*}&\left( \theta_1 - \theta_2 \right) \Bigg[ \int_0^1 \varphi' \left( t \theta_1 + (1-t) \theta_2 \right) \\&\qquad \times \psi' \left( t \left( s \frac{\theta_1 +\theta_2}{2} + (1-s) \theta_2 \right) + (1-t) \left( s \frac{\theta_1 +\theta_2}{2} + (1-s) \theta_1 \right) \right) \mathrm{d} t \\&\quad -(1-s) \int_0^1 \varphi \left( t \theta_1 + (1-t) \theta_2 \right) \\&\qquad \times \psi" \left( t \left( s \frac{\theta_1 +\theta_2}{2} + (1-s) \theta_2 \right) + (1-t) \left( s \frac{\theta_1 +\theta_2}{2} + (1-s) \theta_1 \right) \right) \mathrm{d} t \Bigg] .\end{align*} $$
Estimating
$$\begin{align*}\left\vert \int_0^1 \left( \int_0^1 \varphi' \psi' \, \mathrm{d} t -(1-s) \int_0^1 \varphi \psi" \, \mathrm{d} t \right) \mathrm{d} s \right\vert\leq \left\Vert \varphi' \right\Vert_\infty \left\Vert \psi' \right\Vert_\infty + \frac{{\| \varphi \|}_\infty \left\Vert \psi" \right\Vert_\infty}{2} , \end{align*}$$
we obtain the conclusion.
Corollary 2.5 Let
$I \subset \mathbb {R}$
be an open interval, and let
$\varphi $
and
$\psi \in C^2 (I)$
. For any
$( \theta _1 , \theta _2 ) \in I \times I$
, we have
$$ \begin{align*} & \left\vert \left( \varphi \left( \theta_1 \right) + \varphi \left( \theta_2 \right) \right) \psi \left( \frac{\theta_1 + \theta_2}{2} \right) - \cos \frac{\theta_1 - \theta_2}{2} \left( \varphi \left( \theta_1 \right) \psi \left( \theta_2 \right) + \varphi \left( \theta_2 \right) \psi \left( \theta_1 \right) \right) \right\vert \\&\quad \leq \left( 2 \left\Vert \varphi' \right\Vert_\infty \left\Vert \psi' \right\Vert_\infty + \left\Vert \varphi \right\Vert_\infty \left\Vert \psi" \right\Vert_\infty + {\| \varphi \|}_\infty {\| \psi \|}_\infty \right) \frac{\left( \theta_1 -\theta_2 \right)^2}{4} . \end{align*} $$
Proof By Taylor’s theorem with integral remainder, we have
$$\begin{align*}\cos \theta = 1 + \int_0^{\theta} ( \xi - \theta ) \cos \xi \, \mathrm{d} \xi. \end{align*}$$
Thus, the left-hand side is
$$ \begin{align*}&\left\vert \left( \varphi \left( \theta_1 \right) + \varphi \left( \theta_2 \right) \right) \psi \left( \frac{\theta_1 + \theta_2}{2} \right) - \varphi \left( \theta_1 \right) \psi \left( \theta_2 \right) - \varphi \left( \theta_2 \right) \psi \left( \theta_1 \right) \right. \\&\quad \left. - \left( \varphi \left( \theta_1 \right) \psi \left( \theta_2 \right) + \varphi \left( \theta_2 \right) \psi \left( \theta_1 \right) \right) \int_0^{\frac{\theta_1 - \theta_2}{2}} \left( \xi - \frac{\theta_1 - \theta_2}{2} \right) \cos \xi \, \mathrm{d} \xi \right\vert .\end{align*} $$
Estimating
$$ \begin{align*} &\left\vert \left( \varphi \left( \theta_1 \right) \psi \left( \theta_2 \right) + \varphi \left( \theta_2 \right) \psi \left( \theta_1 \right) \right) \int_0^{\frac{\theta_1 - \theta_2}{2}} \left( \xi - \frac{\theta_1 - \theta_2}{2} \right) \cos \xi \, \mathrm{d} \xi \right\vert\\ &\quad \leq 2 {\| \varphi \|}_\infty {\| \psi \|}_\infty \int_0^{\frac{\theta_1 - \theta_2}{2}} \left( \frac{\theta_1 - \theta_2}{2} - \xi \right) \, \mathrm{d} \xi \\&\quad = {\| \varphi \|}_\infty {\| \psi \|}_\infty \frac{\left( \theta_1- \theta_2 \right)^2}{4} , \end{align*} $$
Lemma 2.4 completes the proof.
Lemma 2.6 Let f and
$g \in C^2 (S^{n-1})$
. For any
$(u_1, u_2 ) \in S^{n-1} \times S^{n-1}$
, we have
$$ \begin{align*} &\left( f \left( u_1 \right) + f \left( u_2 \right) \right) g \left( u_3 \right) - \cos \frac{\angle \left( u_1 ,u_2 \right)}{2} \left( f \left( u_1 \right) g \left( u_2 \right) + f \left( u_2 \right) g \left( u_1 \right) \right) \\&\quad \geq - \Phi_0 (f,g) \frac{\angle \left( u_1 ,u_2 \right)^2}{4} .\end{align*} $$
Proof If
$f =g$
, then the left-hand side is
$\Gamma [f] (u_1, u_2)$
. Thus, in the case (II-iii) of the definition of
$\Phi _0$
, the proof is completed. Let us consider the other cases.
There exists a triple
$(q , \theta _1 , \theta _2 ) \in SO (n) \times \mathbb {R} \times \mathbb {R}$
such that
$u_j = q \phi ( \theta _j , 0 , \ldots , 0 )$
. Then, we have
$u_3 = q \phi ( ( \theta _1 + \theta _2 ) /2 , 0 , \ldots , 0 )$
. Put
$\varphi _q ( \theta ) = f ( q \phi ( \theta , 0 ,\ldots , 0 ) )$
and
$\psi _q (\theta ) = g ( q \phi ( \theta , 0 ,\ldots , 0 ) )$
. In the case (I-i) or (II-i), Corollary 2.5 with
$(\varphi ,\psi ) = ( \varphi _q , \psi _q )$
completes the proof. In the case (I-ii) or (II-ii), the inequality
$\cos \leq 1$
and Lemma 2.4 with
$(\varphi ,\psi ) = (\varphi _q , \psi _q)$
complete the proof.
Corollary 2.7 Let f and
$g \in C^2 (S^{n-1})$
. For any
$(u_1, u_2 ) \in S^{n-1} \times S^{n-1}$
, we have
$$\begin{align*}\Delta [f, g] \left( u_1 ,u_2 \right) \geq - \left( \Phi_0 (f,g) + \Phi_0 (g,f) \right) \frac{\angle \left( u_1 ,u_2 \right)^2}{4}. \end{align*}$$
Corollary 2.8 Let
$f \in C^2 (S^{n-1})$
. For any
$(u_1 , u_2 ) \in S^{n-1}$
, we have
$$\begin{align*}\Gamma [f] \left( u_1, u_2 \right) \geq - \Phi_0 (f,f) \frac{\angle \left( u_1 , u_2 \right)^2}{4}. \end{align*}$$
Lemma 2.9 Let
$I \subset \mathbb {R}$
be an open interval, and let
$\varphi \in C^2 (I)$
. For any
$( \theta _1 , \theta _2 ) \in I \times I$
, we have
$$\begin{align*}\left\vert 2 \varphi \left( \frac{\theta_1 + \theta_2}{2} \right) - \varphi \left( \theta_1 \right) - \varphi \left( \theta_2 \right) \right\vert\leq \left\Vert \varphi" \right\Vert_\infty \frac{\left( \theta_1 -\theta_2 \right)^2}{4} . \end{align*}$$
Proof Lemma 2.4 with
$(\varphi , \psi ) = (1, \varphi )$
completes the proof.
Corollary 2.10 Let
$I \subset \mathbb {R}$
be an open interval, and let
$\varphi \in C^2 (I)$
. For any
$( \theta _1 , \theta _2 ) \in I \times I$
, we have
$$ \begin{align*} &\left\vert 2 \varphi \left( \frac{\theta_1 + \theta_2}{2} \right) + \left( 1- 2 \cos \frac{\theta_1 - \theta_2}{2} \right) \left( \varphi \left( \theta_1 \right) + \varphi \left( \theta_2 \right) \right) \right\vert \\&\quad \leq \left( 2 {\| \varphi \|}_\infty + \left\Vert \varphi" \right\Vert_\infty \right) \frac{\left( \theta_1 -\theta_2 \right)^2}{4} . \end{align*} $$
Proof Applying Taylor’s theorem with integral remainder to
$\cos $
, in the same manner as in Corollary 2.5, Lemma 2.9 completes the proof.
Remark 2.11 Let f be a function defined on
$S^{n-1}$
, and let
$c \in \mathbb {R}$
. For any
$(u_1 , u_2 ) \in S^{n-1} \times S^{n-1}$
, we have
$$ \begin{align*} \Delta [f,c ] \left( u_1 ,u_2 \right) &= \Delta [f,1 ] \left( u_1 , u_2 \right) c \\ &= \Delta [cf ,1 ] \left( u_1, u_2 \right) \\ &= \left( 2 f \left( u_3 \right) + \left( 1- \left\vert u_1 + u_2 \right\vert \right) \left( f \left( u_1 \right) + f \left( u_2 \right) \right) \right) c \\ &= \left( 2 f \left( u_3 \right) + \left( 1- 2 \cos \frac{\angle \left( u_1, u_2 \right)}{2} \right) \left( f \left( u_1 \right) + f \left( u_2 \right) \right) \right) c. \end{align*} $$
Lemma 2.12 Let
$f \in C^2 (S^{n-1})$
, and let
$c \in \mathbb {R}$
. For any
$(u_1, u_2 ) \in S^{n-1} \times S^{n-1}$
, we have
$$\begin{align*}\Delta [f ,c] \left( u_1 , u_2 \right) \geq - \Phi_1 (f) \vert c \vert \frac{\angle \left( u_1, u_2 \right)^2}{4}. \end{align*}$$
Proof Fix an arbitrary
$(u_1, u_2 ) \in S^{n-1} \times S^{n-1}$
. There exists a triple
$(q , \theta _1 , \theta _2 ) \in SO (n) \times \mathbb {R} \times \mathbb {R}$
such that
$u_j = q \phi ( \theta _j , 0 , \ldots , 0 )$
. Then, we have
$u_3 = q \phi ( ( \theta _1 + \theta _2 ) / 2 , 0 , \ldots , 0 )$
. Put
$\varphi _q ( \theta ) = f (q \phi ( \theta , 0 ,\ldots , 0 ) )$
. If
$c f (u) <0$
for some
$u \in S^{n-1}$
, then Remark 2.11 and Corollary 2.10 with
$\varphi = \varphi _q$
complete the proof. If
$c f(u) \geq 0$
for any
$u \in S^{n-1}$
, then the triangle inequality
$\vert u_1 + u_2 \vert \leq 2$
and Lemma 2.9 with
$\varphi = \varphi _q$
complete the proof.
Remark 2.13 Let f and g be functions defined on
$S^{n-1}$
. We have
$\Gamma [f+g] = \Gamma [f] + \Delta [f,g] + \Gamma [g]$
.
Remark 2.14 By Taylor’s theorem with integral remainder, for any
$\theta \in \mathbb {R}$
, we have
$$\begin{align*}\sin^2 \theta = \frac{1-\cos 2 \theta}{2} = \theta^2 + \frac{1}{12} \int_0^{2 \theta} ( \xi -2\theta )^3 \cos \xi \, \mathrm{d} \xi \geq \theta^2 - \frac{\theta^4}{3}. \end{align*}$$
Remark 2.15 Let
$c \in \mathbb {R}$
. By Remark 2.14, for any
$(u_1, u_2 ) \in S^{n-1} \times S^{n-1}$
, we have
$$\begin{align*}\Gamma [ c] \left( u_1 ,u_2 \right) = 4 c^2 \sin^2 \frac{\angle \left( u_1, u_2 \right)}{4} \geq \left( 1-\frac{\pi^2}{48} \right) c^2 \frac{\angle \left( u_1, u_2 \right)^2}{4}. \end{align*}$$
Corollary 2.16 Let
$f \in C^2 (S^{n-1} )$
, and let
$c \in \mathbb {R}$
. For any
$(u_1, u_2 ) \in S^{n-1} \times S^{n-1}$
, we have
$$\begin{align*}\Gamma [ f + c ] \left( u_1, u_2 \right) \geq \left( \left( 1 - \frac{\pi^2}{48} \right) c^2 - \Phi_1 (f) \vert c \vert - \Phi_0 ( f ,f ) \right) \frac{\angle \left( u_1, u_2 \right)^2}{4}. \end{align*}$$
Proof By Remarks 2.13 with
$g =c$
and 2.15, we have
$$\begin{align*}\Gamma [f+c] \left( u_1, u_2 \right) = \Gamma [f] \left( u_1 , u_2 \right) + \Delta [ f,c ] \left( u_1, u_2 \right) + 4 c^2 \sin^2 \frac{\angle \left( u_1 , u_2 \right)}{4}. \end{align*}$$
Applying Corollary 2.8, Lemma 2.12, and the inequality in Remark 2.15 to the first, second, and third terms, respectively, we obtain the conclusion.
3 Main results
3.1 Construction of convex radial sums
We keep the notation from the Introduction and Section 2.1.
Proposition 3.1 Let
$\gamma \in [0,+\infty )$
, and let
$K \in \mathcal {K}^n_0$
be such that
$\rho _K \in C^2 (S^{n-1})$
. Assume that, for any
$( u_1 , u_2 ) \in S^{n-1} \times S^{n-1}$
, the inequality
$\Gamma [ \rho _K ] ( u_1 ,u_2 ) \geq \gamma \angle ( u_1 , u_2 )^2$
holds. Let
$f \in C^2 ( S^{n-1} )$
be such that
$\rho _K + f \geq 0$
. If the inequality
$$\begin{align*}4 \gamma \geq \Phi_0 ( f ,f ) + \Phi_0 \left( f , \rho_K \right) + \Phi_0 \left( \rho_K , f \right) \end{align*}$$
holds, then
$K_f$
is convex.
Proof Let us check the condition (ii) in Lemma 2.1. Let
$( u_1 , u_2 ) \in S^{n-1} \times S^{n-1}$
be such that
$u_1 + u_2 \neq 0$
. By Remark 2.13 with
$g = \rho _K$
and the assumption, we have
$$\begin{align*}\Gamma \left[ \rho_K + f \right] = \Gamma \left[ f + \rho_K \right] \geq \Gamma [ f ] \left( u_1, u_2 \right) + \Delta \left[ f, \rho_K \right] \left( u_1, u_2 \right) + \gamma \angle \left( u_1, u_2 \right)^2. \end{align*}$$
Corollaries 2.7 with
$g = \rho _K$
and 2.8 complete the proof.
Theorem 3.3 Let
$A \in \mathcal {S}^n$
be such that
$\rho _A \in C^2 (S^{n-1} )$
. Let
$\gamma \in [0,+\infty )$
, and put
$$\begin{align*}R_A (\gamma) = \frac{24}{48-\pi^2} \left( \Phi_1 \left( \rho_A \right) + \sqrt{ \Phi_1 \left( \rho_A \right) ^2 + \frac{48-\pi^2}{12} \left( \Phi_0 \left( \rho_A , \rho_A \right) + 4 \gamma \right)} \right). \end{align*}$$
If
$r \geq R_A (\gamma )$
, then, for any
$( u_1, u_2 ) \in S^{n-1} \times S^{n-1}$
, we have
$$\begin{align*}\Gamma \left[ \rho_A + r \right] \left( u_1, u_2 \right) \geq \gamma \angle \left( u_1, u_2 \right)^2. \end{align*}$$
Proof This follows from Corollary 2.16 with
$(f,c) = ( \rho _A ,r )$
.
Corollary 3.4 Let A and
$R_A$
be as in Theorem 3.3. If
$r \geq R_A (0)$
, then
$A \mathop{\widetilde{+}} r B^n$
is convex.
Combining Theorem 3.3 and Proposition 3.1, for any
$A \in \mathcal {S}^n$
with
$\rho _A \in C^2 ( S^{n-1})$
, there exists a “large enough”
$F \in C^2 (S^{n-1})$
such that
$A_F$
is convex. Precisely, we obtain the following corollary.
Corollary 3.5 Let A,
$\gamma $
, and
$R_A$
be as in Theorem 3.3. Let
$( r ,f) \in [ R_A (\gamma ) , +\infty ) \times C^2 (S^{n-1})$
be such that
$\rho _A +r+f \geq 0$
. If the inequality
$$\begin{align*}4 \gamma \geq \Phi_0 ( f , f) + \Phi_0 \left( f, \rho_A +r \right) + \Phi_0 \left( \rho_A +r ,f \right) \end{align*}$$
holds, then
$A_{r+f}$
is convex.
Proof Put
$K = A \mathop{\widetilde{+}} rB^n$
.
$\rho _K = \rho _A +r$
is of class
$C^2$
. Theorem 3.3 guarantees that, for any
$(u_1, u_2 ) \in S^{n-1} \times S^{n-1}$
, the inequality
$\Gamma [ \rho _K ] ( u_1, u_2 ) \geq \gamma \angle ( u_1 ,u_2)^2$
holds. By Proposition 3.1,
$A_{r+f} = K_f$
is convex.
3.2 Construction of convex intersection bodies
We keep the notation from the Introduction and Section 2.1.
Remark 3.6 Let
$A \in \mathcal {S}^n$
. Let
$f \in C^0 ( S^{n-1})$
be such that
$\rho _A + f \geq 0$
. For any
$u \in S^{n-1}$
, we have
$$ \begin{align*} V_{n-1} \left( A_f \cap u^\perp \right) &= \frac{1}{n-1} \mathcal{R} \left[ \left( \rho_A +f \right)^{n-1} \right] (u) \\ &= \frac{1}{n-1} \sum_{i=0}^{n-1} \binom{n-1}{i} \left\langle \rho_A^i , f^{n-1-i} \right\rangle ( u) \\ &= V_{n-1} \left( A \cap u^\perp \right) + \frac{1}{n-1} \sum_{i=0}^{n-2} \binom{n-1}{i} \left\langle \rho_A^i , f^{n-1-i} \right\rangle ( u) \\ &=\frac{1}{n-1} \mathcal{R} \left[ f^{n-1} \right] (u) + \frac{1}{n-1} \sum_{i=1}^{n-1} \binom{n-1}{i} \left\langle \rho_A^i , f^{n-1-i} \right\rangle ( u). \end{align*} $$
In particular, if f is a constant c, then
$$ \begin{align*} V_{n-1} \left( A_c \cap u^\perp \right) &= V_{n-1} \left( A \cap u^\perp \right) + \frac{1}{n-1} \sum_{i=0}^{n-2} \binom{n-1}{i} \mathcal{R} \left[ \rho_A^i \right] (u) c^{n-1-i} \\ &=\kappa_{n-1} c^{n-1} + \frac{1}{n-1} \sum_{i=1}^{n-1} \binom{n-1}{i} \mathcal{R} \left[ \rho_A^i \right] (u) c^{n-1-i}. \end{align*} $$
Proposition 3.7 Let
$\gamma \in [0,+\infty )$
, and let
$A \in \mathcal {S}^n$
be such that
$\rho _A \in C^2 ( S^{n-1})$
. Assume that, for any
$(u_1, u_2 ) \in S^{n-1} \times S^{n-1}$
, the inequality
$\Gamma [ \rho _{IA} ] (u_1, u_2 ) \geq \gamma \angle (u_1, u_2)^2$
holds. Let
$f \in C^2 (S^{n-1})$
be such that
$\rho _A + f \geq 0$
. If the inequality
$$ \begin{align*} 4\gamma (n-1)^2&\geq\Phi_0 \left( \mathcal{R} \left[ \rho_A^{n-1} \right] , \sum_{i=0}^{n-2} \left\langle \rho_A^i , f^{n-1-i} \right\rangle \right) \\&\quad + \Phi_0 \left( \sum_{i=0}^{n-2} \left\langle \rho_A^i , f^{n-1-i} \right\rangle , \mathcal{R} \left[ \rho_A^{n-1} \right] \right) \\&\quad + \Phi_0 \left( \sum_{i=0}^{n-2} \left\langle \rho_A^i , f^{n-1-i} \right\rangle , \sum_{i=0}^{n-2} \left\langle \rho_A^i , f^{n-1-i} \right\rangle \right) \end{align*} $$
holds, then
$I A_f$
is convex.
Proof Let
$(u_1, u_2 ) \in S^{n-1} \times S^{n-1}$
be such that
$u_1 + u_2 \neq 0$
. Let us check the condition (ii) in Lemma 2.1. By Remarks 3.6 and 2.13, we have
$$ \begin{align*} \Gamma \left[ \rho_{IA_f} \right]&= \Gamma \left[ \rho_{IA} \right]+ \Delta \left[ \frac{1}{n-1} \mathcal{R} \left[ \rho_A^{n-1} \right] , \frac{1}{n-1} \sum_{i=0}^{n-2} \left\langle \rho_A^i , f^{n-1-i} \right\rangle \right] \\&\quad+ \Gamma \left[ \frac{1}{n-1} \sum_{i=0}^{n-2} \left\langle \rho_A^i , f^{n-1-i} \right\rangle \right] .\end{align*} $$
By the assumption, we have
$\Gamma [ \rho _{IA} ] (u_1, u_2 ) \geq \gamma \angle (u_1, u_2 )^2$
. Hence, Corollaries 2.7 and 2.8 complete the proof.
Remark 3.8 Let
$A \in \mathcal {S}^n$
be such that
$\rho _A \in C^2 (S^{n-1})$
, and let
$\gamma \in [ 0 , + \infty )$
. The function of
$r \in [0,+\infty )$
,
$$ \begin{align*} &\left( 1 - \frac{\pi^2}{48} \right) \kappa_{n-1}^2 r^{2(n-1)} - \frac{\kappa_{n-1}}{n-1} \Phi_1 \left( \sum_{i=1}^{n-1} \binom{n-1}{i} \mathcal{R} \left[ \rho_A^i \right] r^{2(n-1)-i} \right) \\ & \quad - \frac{1}{(n-1)^2} \Phi_0 \left( \sum_{i=1}^{n-1} \binom{n-1}{i} \mathcal{R} \left[ \rho_A^i \right] r^{n-1-i} , \sum_{i=1}^{n-1} \binom{n-1}{i} \mathcal{R} \left[ \rho_A^i \right] r^{n-1-i} \right) -4 \gamma , \end{align*} $$
has at least one real root, and the set of the real roots is bounded. In fact, we have the following asymptotic behaviors as r goes to infinity:
$$ \begin{align*}&\Phi_1 \left( \sum_{i=1}^{n-1} \binom{n-1}{i} \mathcal{R} \left[ \rho_A^i \right] r^{2(n-1)-i} \right)\leq \sum_{i=1}^{n-1} \binom{n-1}{i} \left\Vert \left( \mathcal{R} \left[ \rho_A^i \right] \circ \phi \right)" \right\Vert_\infty r^{2(n-1)-i} = O \left( r^{2n-3} \right) \!, \\&\Phi_0 \left( \sum_{i=1}^{n-1} \binom{n-1}{i} \mathcal{R} \left[ \rho_A^i \right] r^{n-1-i} , \sum_{i=1}^{n-1} \binom{n-1}{i} \mathcal{R} \left[ \rho_A^i \right] r^{n-1-i} \right) \\&\quad\leq \sum_{i=1}^{n-1} \sum_{j=1}^{n-1} \binom{n-1}{i} \binom{n-1}{j} \left( 2 \left\Vert \left( \mathcal{R} \left[ \rho_A^i \right] \circ \phi \right)' \right\Vert_\infty \left\Vert \left( \mathcal{R} \left[ \rho_A^j \right] \circ \phi \right)' \right\Vert_\infty \right. \\&\qquad \left. + \left\Vert \mathcal{R} \left[ \rho_A^i \right] \circ \phi \right\Vert_\infty \left\Vert \left( \mathcal{R} \left[ \rho_A^j \right] \circ \phi \right)" \right\Vert_\infty \right) r^{2(n-1)-i-j} \\ & \quad = O \left( r^{2n-4} \right) . \end{align*} $$
Theorem 3.9 Let
$A \in \mathcal {S}^n$
be such that
$\rho _A \in C^2 (S^{n-1})$
, and let
$\gamma \in [ 0 , + \infty )$
. Let
$\widetilde {R}_A (\gamma )$
be the maximum real root of the function of r,
$$ \begin{align*} &\left( 1 - \frac{\pi^2}{48} \right) \kappa_{n-1}^2 r^{2(n-1)} - \frac{\kappa_{n-1}}{n-1} \Phi_1 \left( \sum_{i=1}^{n-1} \binom{n-1}{i} \mathcal{R} \left[ \rho_A^i \right] r^{2(n-1)-i} \right) \\&\quad- \frac{1}{(n-1)^2} \Phi_0 \left( \sum_{i=1}^{n-1} \binom{n-1}{i} \mathcal{R} \left[ \rho_A^i \right] r^{n-1-i} , \sum_{i=1}^{n-1} \binom{n-1}{i} \mathcal{R} \left[ \rho_A^i \right] r^{n-1-i} \right) -4 \gamma. \end{align*} $$
If
$r \geq \widetilde {R}_A (\gamma )$
, then, for any
$(u_1, u_2 ) \in S^{n-1} \times S^{n-1}$
, we have
$$\begin{align*}\Gamma \left[ \rho_{I(A \mathop{\widetilde{+}} rB^n)} \right] \left( u_1, u_2 \right) \geq \gamma \angle \left( u_1, u_2 \right)^2. \end{align*}$$
Proof Remark 3.6 with
$c=r$
and Corollary 2.16 with
$$\begin{align*}(f,c)= \left( \frac{1}{n-1} \sum_{i=1}^{n-1} \binom{n-1}{i} \mathcal{R} \left[ \rho_A^i \right] r^{n-1-i} , \kappa_{n-1} r^{n-1} \right) \end{align*}$$
complete the proof.
Corollary 3.10 Let A and
$\widetilde {R}_A$
be as in Theorem 3.9. If
$r \geq \widetilde {R}_A (0)$
, then
$I ( A \mathop{\widetilde{+}} rB^n)$
is convex.
Combining Theorem 3.9 and Proposition 3.7, for any
$A \in \mathcal {S}^n$
with
$\rho _A \in C^2 ( S^{n-1})$
, there exists a “large enough”
$F \in C^2 (S^{n-1})$
such that
$I A_F$
is convex. Precisely, we obtain the following corollary.
Corollary 3.11 Let A,
$\gamma $
, and
$\widetilde {R}_A$
be as in Theorem 3.9. Let
$( r ,f) \in [\widetilde {R}_A (\gamma ) , +\infty ) \times C^2 (S^{n-1})$
be such that
$\rho _A +r+f \geq 0$
. If the inequality
$$ \begin{align*} 4\gamma (n-1)^2 &\geq \Phi_0 \left( \mathcal{R} \left[ \left( \rho_A +r \right)^{n-1} \right] , \sum_{i=0}^{n-2} \left\langle \left( \rho_A +r \right)^i , f^{n-1-i} \right\rangle \right) \\ &\quad + \Phi_0 \left( \sum_{i=0}^{n-2} \left\langle \left( \rho_A +r \right)^i , f^{n-1-i} \right\rangle , \mathcal{R} \left[ \left( \rho_A +r \right)^{n-1} \right] \right) \\ &\quad + \Phi_0 \left( \sum_{i=0}^{n-2} \left\langle \left( \rho_A +r \right)^i , f^{n-1-i} \right\rangle , \sum_{i=0}^{n-2} \left\langle \left( \rho_A +r \right)^i , f^{n-1-i} \right\rangle \right) \end{align*} $$
holds, then
$I A_{r+f}$
is convex.
Acknowledgment
The author would like to express his deep gratitude to the anonymous reviewer(s) for careful reading of this paper and helpful suggestions.
