2.1. The joint distribution of $({\boldsymbol\ell}^{(n)}[\kappa],\mathbf{s}^{(n)}[\kappa])$

As announced in the introduction, in this section we will consider polygons $K\in\mathbf{P}^{\mathcal{T}}$ , that is, such that the set $\mathsf{Dom}(K)$ is tangent to all sides of K.

Lemma 2.1. For all $K\in\mathbf{P}^{\mathcal{T}},$

\[\mathrm{P}_K(n)\underset{n\to+\infty}{\sim}\widetilde{\mathrm{P}}_K(n).\]

Proof. The proof is similar to that of the $K=\mathfrak{C}_\kappa$ case. For an n-tuple $\mathbf{z}[n]$ under $\mathsf{U}^{(n)}_{K}$ , Bárány’s theorem (Theorem 2.1) immediately implies that

\[\mathrm{P}\big(\mathsf{PCP}(\mathbf{z}[n])\text{ is `full-sided'}\ \vert\ \mathbf{z}[n]\in\mathcal{C}_K(n)\big)=\frac{\widetilde{\mathrm{P}}_K(n)}{\mathrm{P}_K(n)}\underset{n\to+\infty}{\longrightarrow}1.\]

We have $\mathsf{Dom}(K)\subset K$ . Hence we can define the $\mathsf{PCP}$ of $\mathsf{Dom}(K)$ as the minimum parallel polygon to K that contains $\mathsf{Dom}(K)$ . Since $K\in\mathbf{P}^{\mathcal{T}}$ , $\mathsf{Dom}(K)$ is tangent at all sides of K, which means that $\mathsf{PCP}(\mathsf{Dom}(K))=K$ . Since the $\mathsf{PCP}$ is a continuous map in the plane, we have

\[d_{\mathrm{H}}(\mathbf{z}[n],\mathsf{Dom}(K))\underset{n\to\infty}{\longrightarrow}0 \implies d_{\mathrm{H}}(\mathsf{PCP}(\mathbf{z}[n]),K)=d_{\mathrm{H}}(\mathsf{PCP}(\mathbf{z}[n]),\mathsf{PCP}(\mathsf{Dom}(K)))\underset{n\to\infty}{\longrightarrow}0.\]

Hence, when n is large, the $\mathsf{PCP}$ has a.s. as many sides as K.

The next lemma shows that the data of $\mathbf{z}[n]$ with ‘full-sided’ $\mathsf{PCP}$ s is sufficient for our search of $\mathrm{P}_K(n)$ . Therefore, a good characterization of this quantity is required.

Notation. In accordance with this lemma, we define the distribution ${\mathsf{D}}^{(n)}_{K}$ that denotes the law of an n-tuple $\mathbf{z}[n]$ taken under $\mathsf{Q}^{(n)}_{K}$ , conditioned to have $s[\kappa](\mathbf{z}[n])\in\mathbb{N}_\kappa(n)$ , that is, conditioned to have a full-sided $\mathsf{PCP}$ . In particular, with $d_V$ the total variation distance, by Lemma 2.1 we have

\[d_V\big(\mathsf{Q}^{(n)}_{K},{\mathsf{D}}^{(n)}_{K}\big)\underset{n\to\infty}{\longrightarrow} 0.\]

For such a $\mathbf{z}[n]$ taken under ${\mathsf{D}}^{(n)}_{K},$ we give in the following theorem the joint distribution of $({\boldsymbol\ell}^{(n)}[\kappa]=\ell[\kappa](\mathbf{z}[n]),\mathbf{s}^{(n)}[\kappa]=s[\kappa](\mathbf{z}[n]))$ , which is analogous to [Reference Morin14, Theorem 3.6].

Theorem 2.2. Let $K\in\mathbf{P}_{\kappa}^{\mathcal{T}}$ , let $\mathbf{z}[n]$ be an n-tuple of random points under ${\mathsf{D}}^{(n)}_{K}$ , and consider the random variables ${\boldsymbol\ell}^{(n)}[\kappa]=\ell[\kappa](\mathbf{z}[n])$ , $\mathbf{s}^{(n)}[\kappa]=s[\kappa](\mathbf{z}[n]).$ Then, for a given $s[\kappa]\in\mathbb{N}_\kappa(n)$ , the pair $({\boldsymbol\ell}^{(n)}[\kappa],\mathbf{s}^{(n)}[\kappa])$ has the joint distribution

(20) \begin{align} \mathrm{P}\big({\boldsymbol\ell}^{(n)}[\kappa]\in\mathrm{d}\ell[\kappa],\mathbf{s}^{(n)}[\kappa]=s[\kappa]\big)=f^{(n)}_{K}(\ell[\kappa],s[\kappa])\mathrm{d}\ell[\kappa], \end{align}

where we denote $\mathrm{d}\ell[\kappa]=\prod_{j=1}^\kappa\mathrm{d}\ell_j$ , and where

(21) \begin{align} f^{(n)}_{K}(\ell[\kappa],s[\kappa])=\frac{n!}{\widetilde{\mathrm{P}}_K(n)} \mathbb{1}_{\ell[\kappa]\in\mathcal{L}_K} \prod_{j=1}^{\kappa}\frac{\sin\!(\theta_j)^{s_j-1}c_j^{s_{{j-1}}+s_j-1}}{s_j!(s_{{j-1}}+s_j-1)!}. \end{align}

Recall that, by (16), $c[\kappa]$ is indeed a function of $\ell[\kappa].$

Recall $\mathcal{C}_K(n)$ defined on page 5, and let us rewrite $f^{(n)}_{K}$ as

\[f^{(n)}_{K}(\ell[\kappa],s[\kappa])\mathrm{d}\ell[\kappa]=\frac{\mathsf{Leb}_{2n}(\{{z}[n]\in\mathcal{C}_K(n)\text{ s.t. }s[\kappa]({z}[n])=s[\kappa],\ell[\kappa]({z}[n])\in\mathrm{d}\ell[\kappa]\})}{\widetilde{\mathrm{P}}_K(n)}.\]

Let us compute the numerator of this expression. Conditional on a $\mathsf{PCP}$ with side-lengths $c[\kappa]$ and its corners, z[n] is in convex position if and only if, for all $j\in\{1,\ldots,\kappa\}$ , the $s_j-1$ points in the jth corner form a convex chain, together with the contact points (this is proved in [Reference Morin14, Lemma 2.3]).

Assume now, for every $j\in\{1,\ldots,\kappa\}$ , that $\mathsf{cp}_j=\mathsf{b}_j+x_j(\mathsf{b}_{{j+1}}-\mathsf{b}_j)$ , that is, positioned at the coordinate $x_jc_j$ on the jth side of the $\mathsf{PCP}$ , with $x_j\in[0,\,1]$ . Then the area of $\mathsf{corner}_j$ (seen as a triangle) is $(1-x_j)c_j x_{{j+1}}c_{{j+1}}\sin\!(\theta_j)/2$ .

Hence, recalling (15) for the bi-pointed triangle, the Lebesgue measure of convex chains in $\mathsf{corner}_j$ is

\[\frac{2^{s_j-1}}{(s_j-1)!s_j!}\Big(\frac{(1-x_j)c_j x_{{j+1}}c_{{j+1}}\sin\!(\theta_j)}{2}\Big)^{s_j-1}.\]

There are $\binom{n}{s_1,\ldots,s_\kappa}$ ways of choosing the vertices in the corners so as to respect the size-vector $s[\kappa]$ . We now have to integrate on the position of the contact points on their respective sides:

\begin{multline*} \widetilde{\mathrm{P}}_K(n)\cdot f^{(n)}_{K}(\ell[\kappa],s[\kappa]) \\ =\binom{n}{s_1,\ldots,s_\kappa}\int_{0}^{1}\cdots\int_{0}^{1}\Bigg[\prod_{j=1}^{\kappa} \left({(1-x_j)c_j x_{{j+1}}c_{{j+1}}\sin\!(\theta_j)}\right)^{s_j-1}\frac{c_j}{(s_j-1)!s_j!}\mathrm{d}x_j\Bigg]. \end{multline*}

With the classical integral identity (relative to beta distributions)

\[ \displaystyle \int_{0}^{1}x^a(1-x)^b\mathrm{d}x=\frac{a!b!}{(a+b+1)!}, \]

this becomes

\begin{align*} \widetilde{\mathrm{P}}_K(n)\cdot f^{(n)}_{K}(\ell[\kappa],s[\kappa])&= n! \prod_{j=1}^{\kappa} \frac{\sin\!(\theta_j)^{s_j-1}c_j^{s_j+s_{{j+1}}-1}}{(s_j+s_{{j+1}}-1)!s_j!}. \end{align*}

2.2. The quantity $\mathrm{AP}^*(K)$ in $f^{(n)}_K$

Bárány’s logarithmic equivalent of $\mathrm{P}_\kappa(n)$ is expressed in terms of the quantity $\mathrm{AP}^*(K)$ . The next step of our strategy consists in understanding how to retrieve this quantity $\mathrm{AP}^*(K)$ in $f^{(n)}_K$ .

We want to describe the asymptotic behaviour of the measure computed in Theorem 2.2. To do so, we will make a change of variables to make apparent the asymptotics of $f^{(n)}_K$ (that is, we want to express $f^{(n)}_{K}$ in terms of simple distributions) so that, eventually, its convergence will appear as a consequence of standard considerations. Let us rewrite

(22) \begin{align} f^{(n)}_{K}(\ell[\kappa],s[\kappa])=\frac{n!}{\widetilde{\mathrm{P}}_K(n)\prod_{j=1}^\kappa \sin\!(\theta_j)r_j}\cdot\Pi\cdot\mathbb{1}_{\ell[\kappa]\in\mathcal{L}_K}, \end{align}

where

\[ \displaystyle\Pi \,:\!= \prod_{j=1}^{\kappa}\frac{r_j\cdot\sin\!(\theta_j)^{s_j}c_j^{s_{{j-1}}+s_j-1}}{s_j!(s_{{j-1}}+s_j-1)!}. \]

By (16), the quantity $\Pi$ may be rewritten as

(23) \begin{align} \Pi= \Bigg[\prod_{j=1}^{\kappa}\frac{\left(r_jr_{{j+1}}\sin\!(\theta_j)\right)^{s_j}}{s_j!(s_{{j-1}}+s_j-1)!}\Bigg]\cdot\prod_{j=1}^{\kappa} \left(1-\frac{\mathfrak{cl}_j(\ell[\kappa])}{r_j}\right)^{s_{{j-1}}+s_j-1}. \end{align}

We will interpret the first product in the right-hand side (rhs) of (23) in terms of Poisson random variables (r.v.) coinciding on $s[\kappa]$ : for two families of positive real numbers $f^{(1)}[\kappa],f^{(2)}[\kappa]$ , set $\bar{f}^{(1)}=\sum_{i=1}^\kappa f^{(1)}_i$ , and let there be $\kappa$ Poisson r.v. $\mathcal{P}^{(n)}_{1,1},\ldots,\mathcal{P}^{(n)}_{1,\kappa}$ of respective parameters $n f^{(1)}_1/\bar{f}^{(1)},\ldots,n f^{(1)}_\kappa/\bar{f}^{(1)}$ , and $\kappa$ other Poisson r.v. $\mathcal{P}^{(n)}_{2,1},\ldots,\mathcal{P}^{(n)}_{2,\kappa}$ of respective parameters $nf^{(2)}_1/\bar{f}^{(1)},\ldots,nf^{(2)}_\kappa/\bar{f}^{(1)}$ , all the r.v. being independent.

For $s[\kappa]\in\mathbb{N}_\kappa(n),$ the probability $\mathbb{M}(s[\kappa])=\mathrm{P} (\mathcal{P}^{(n)}_{1,j}=s_j,\mathcal{P}^{(n)}_{2,j}=s_{{j-1}}+s_j-1,j\in\{1,\ldots,\kappa\} )$ satisfies

(24) \begin{align}\nonumber \mathbb{M}(s[\kappa])&=\prod_{j=1}^{\kappa}\frac{\mathrm{e}^{-nf^{(1)}_j/\bar{f}^{(1)}} \big (nf^{(1)}_j/\bar{f}^{(1)} \big )^{s_j}}{s_j!} \cdot\frac{\mathrm{e}^{-nf^{(2)}_j/\bar{f}^{(1)}}(nf^{(2)}_j/\bar{f}^{(1)})^{s_{{j-1}}+s_j-1}}{(s_{{j-1}}+s_j-1)!}\\ &=\frac{n^{3n-\kappa} \mathrm{e}^{-\lambda}}{(\bar{f}^{(1)} )^{3n-\kappa}\prod_{j=1}^\kappa f^{(2)}_j}\prod_{j=1}^{\kappa}\frac{\big(\,f^{(1)}_jf^{(2)}_{{j+1}}f^{(2)}_j\big)^{s_j}}{s_j!(s_{{j-1}}+s_j-1)!} \end{align}

where $\lambda = ({n} / {\bar{f}^{(1)}}) \sum_{j=1}^{\kappa} (\,f^{(1)}_j+f^{(2)}_j )$ , so that the last term of the equation is very similar to (23) if we choose $(\,f^{(1)}[\kappa],f^{(2)}[\kappa])$ carefully. To fit the numerator in (23) and (24), it suffices to pick $f^{(1)}$ , $f^{(2)}$ so that they solve the following system:

(25) \begin{align} \left\{ \begin{array}{rl}\displaystyle f^{(2)}_j & = f^{(1)}_j+f^{(1)}_{{j-1}} \\[3pt] f^{(1)}_jf^{(2)}_{{j+1}}f^{(2)}_j& = r_j\cdot r_{j+1}\cdot\sin\!(\theta_j),\qquad j\in\{1,\ldots,\kappa\}. \end{array} \right. \end{align}

The variables $f^{(2)}$ can be eliminated from (25), so that we retrieve the parametrizing system $\mathsf{PS}(\theta[\kappa],r[\kappa])$ stated in (4):

\begin{align*} \mathsf{PS}(\theta[\kappa],r[\kappa]):\big\{ \begin{array}{ll}\displaystyle f^{(1)}_j\big(\,f^{(1)}_j+f^{(1)}_{{j-1}}\big)\big(\,f^{(1)}_j+f^{(1)}_{{j+1}}\big) = r_j\cdot r_{j+1}\cdot\sin\!(\theta_j),\qquad j\in\{1,\ldots,\kappa\}. \end{array} \end{align*}

2.2.1. The link with the affine perimeter.

Let us assume for a moment that we have found a positive vector $(\,f[\kappa]\,:\!=f_1[\kappa])$ solution to $\mathsf{PS}$ .

We may now prove Theorem 1.2, which states that, for all $K\in \mathbf{P}_{\kappa}^{\mathcal{T}}$ , $f_j=\sqrt[3]{2T_j},$ for all $j\in\{1,\ldots,\kappa\}$ . To begin with, we start by proving that the supremum over all affine perimeters is given by

(26) \begin{align} \mathrm{AP}^*(K)=2^{2/3}\sum_{i=1}^\kappa f_i. \end{align}

Proof of Theorem 1.2. For any convex set K, the quantity $\mathrm{AP}^*(K)$ is defined as

\[ \displaystyle\mathrm{AP}^*(K)\,:\!=\max_{\substack{S\ \text{convex set} \\ S\subset K}} \mathrm{AP}(S); \]

recall Theorem 1.1. Let $\mathsf{SubTan}_{K}$ be the set of convex sets contained in K, such that their boundary is tangent at all sides of K and that they are composed of parabolic arcs between the tangency points. Since K is in $\mathbf{P}_{\kappa}^{\mathcal{T}}$ , by Bárány’s limit shape theorem, we know that $\mathsf{Dom}(K)\in \mathsf{SubTan}_{K}.$ Therefore, $\mathrm{AP}^*(K)$ may be rewritten as

(27) \begin{align} \mathrm{AP}^*(K)&=\max_{\mathcal{C}\in\mathsf{SubTan}_{K}}\mathrm{AP}(\mathcal{C}). \end{align}

Let $\mathcal{C}\in \mathsf{SubTan}_{K}$ be one of these sets. We denote by $\mathsf{p}_j=\mathsf{p}_j(\mathcal{C})$ the point of tangency of $\mathcal{C}$ with the jth side of K. The affine perimeter of $\mathcal{C}$ is given, by definition, as

(28) \begin{align} \mathrm{AP}(\mathcal{C})=2\sum_{i=1}^\kappa \sqrt[3]{T_i}, \end{align}

where $T_i=T_i(\mathcal{C})$ , $i\in\{1,\ldots,\kappa\}$ , denotes the area of the triangle delineated by the vertices $\mathsf{p}_{i},\mathsf{v}_{{i+1}},\mathsf{p}_{{i+1}}$ (see Figure 8).

Figure 8.

If K is in $\mathbf{P}_{\kappa}^{\mathcal{T}},$ pick $\mathcal{C}\in\mathsf{SubTan}_{K}$ , a curve that is tangent to every side of K at $\mathsf{p}_j$ for the jth side. This curve is not necessarily the boundary of $\mathsf{Dom}(K).$ The triangles marked with dots are the triangles $T_i$ , $i\in\{1,\ldots,\kappa\}$ .

Figure 8 Long description

A diagram of a geometric shape with labeled points, curves, and triangles. The shape is a polygon with vertices labeled as v1, v2, vk, and vj. Points p1, p2, pk, and pj are marked on the sides of the polygon. Curves are drawn from each vertex to the corresponding point on the adjacent sides. The triangles marked with dots are labeled as Ti, where i ranges from 1 to kappa. These triangles are tangent to every side of the polygon at the points pj for the jth side. The curves are not necessarily the boundary of the domain of the polygon.

Set $u_j=u_j(\mathcal{C})\,:\!=\Vert\mathsf{v}_j-\mathsf{p}_j\Vert/r_j$ . With this notation, rewriting the area $T_i$ in terms of the geometric properties of K and $\mathcal{C}$ turns (28) into

(29) \begin{align}\nonumber \mathrm{AP}(\mathcal{C})&=2\sum_{i=1}^\kappa \left(r_iu_i\cdot r_{{i+1}}(1-u_{{i+1}})\cdot \sin\!(\theta_i)/2\right)^{1/3}\\ &=2^{2/3}\psi(r[\kappa],\theta[\kappa],u[\kappa]), \end{align}

where $\psi(r[\kappa],\theta[\kappa],u[\kappa])\,:\!=\sum_{i=1}^\kappa \left(r_iu_i\cdot r_{{i+1}}(1-u_{{i+1}})\cdot \sin\!(\theta_i)\right)^{1/3}$ . Therefore, the quantity $\mathrm{AP}^*(K)$ is, by (27), given by

(30) \begin{align} \mathrm{AP}^*(K)&=2^{2/3}\max_{{u[\kappa]\in[0,\,1]^\kappa}}\psi(r[\kappa],\theta[\kappa],u[\kappa]), \end{align}

since taking the supremum on all $u[\kappa]\left(:\!=u[\kappa](\mathcal{C})\right)$ results in taking the supremum on the $u[\kappa]$ in [0, 1] (and it becomes a maximum, since we know $\mathrm{AP}^*(K)$ is attained for some $\mathcal{C}$ ). By (4), we have

\[\psi(r[\kappa],\theta[\kappa],u[\kappa])=\sum_{i=1}^{\kappa}\left[f_i\left(\,f_i+f_{{i-1}}\right)u_i\left(\,f_i+f_{{i+1}}\right)(1-u_{{i+1}})\right]^{1/3}.\]

By the classical convexity inequality $\displaystyle (abc)^{1/3}\leq (a+b+c)/3$ for all $a,b,c\geq0$ , we easily see that, for all ${u[\kappa]\in[0,\,1]^\kappa}$ , we have

\[\psi(r[\kappa],\theta[\kappa],u[\kappa])\leq\frac1{3}\sum_{i=1}^{\kappa}\left[f_i+\left(\,f_i+f_{{i-1}}\right)u_i+\left(\,f_i+f_{{i+1}}\right)(1-u_{{i+1}})\right]=\sum_{i=1}^{ \kappa}f_i. \]

Hence, (30) yields $\mathrm{AP}^*(K)\leq2^{2/3}\sum_{i=1}^{ \kappa}f_i.$ Now set $ w_j\,:\!= ({f_j})/({f_{{j-1}}+f_j})$ for all $j\in\{1,\ldots,\kappa\}$ . We have $\psi(r[\kappa],\theta[\kappa],w[\kappa])=\sum_{i=1}^{\kappa}f_i$ , which yields (26) and (5).

Now, substituting $w[\kappa]$ in $\mathsf{PS}$ gives

\[f_j^3=w_jr_j(1-w_{{j+1}})r_{{j+1}}\sin\!(\theta_j)=2T_j, \qquad\text{ for all }j\in\{1,\ldots,\kappa\}.\]

Hence, if $f[\kappa]$ is a solution to $\mathsf{PS}(\theta[\kappa],r[\kappa])$ , it is equal to $((2T_1)^{1/3},\ldots,(2T_\kappa)^{1/3})$ , where the quantities $T[\kappa]$ are those of the maximizing curve $\mathcal{C}=\mathsf{Dom}(K)$ , which is unique. This means that a solution to $\mathsf{ PS}(\theta[\kappa],r[\kappa])$ is unique. Conversely, we can check that $(\sqrt[3]{2T_1},\ldots,\sqrt[3]{2T_\kappa})$ is a solution to $\mathsf{PS}$ with such $w[\kappa]$ . This gives Theorem 1.2.

Proof of Theorem 1.3. Set $\bar{f}=\sum_{i=1}^{\kappa}f_i$ and the family $g[\kappa]$ of real numbers, defined for all $j\in\{1,\ldots,\kappa\}$ as

\[g_j\,:\!=f_j/\bar{f}.\]

For the $\lambda$ defined just after (24), we have $\lambda=3n$ , and we rewrite the joint ‘distribution’ $f^{(n)}_K$ as

(31) \begin{align} f^{(n)}_{K}(\ell[\kappa],s[\kappa])=\mathsf{Faktor}(n)\mathbb{1}_{\ell[\kappa]\in\mathcal{L}_K}\mathbb{M}(s[\kappa])\cdot\prod_{j=1}^\kappa\left(1-\frac{\mathfrak{cl}_j(\ell[\kappa])}{r_j}\right)^{s_{{j-1}}+s_j-1}, \end{align}

where

(32) \begin{equation} \mathsf{Faktor}(n)\,:\!=\Bigg(\prod_{j=1}^\kappa\frac{f_j+f_{{j-1}}}{\sin\!(\theta_j)r_j}\Bigg)\cdot\frac{n!\ \mathrm{e}^{3n}(\bar{f})^{3n-\kappa}}{\widetilde{\mathrm{P}}_K(n)\ n^{3n-\kappa}}, \end{equation}

and $\mathbb{M}(s[\kappa])$ was introduced in (24).

We get to the main theorem of this section, which describes the asymptotic behaviour of the renormalized geometric variables of the $\mathsf{PCP}.$

Theorem 2.3. For $\mathbf{z}[n]$ under ${\mathsf{D}}^{(n)}_{K}$ , consider ${\boldsymbol\ell}^{(n)}[\kappa]=\ell[\kappa](\mathbf{z}[n])$ , $\mathbf{s}^{(n)}[\kappa]=s[\kappa](\mathbf{z}[n])$ . We introduce the random variables $\overline{{\boldsymbol\ell}}^{(n)}[\kappa]=n{\boldsymbol\ell}^{(n)}[\kappa]$ and $\displaystyle\mathbf{x}_j^{(n)}=({\mathbf{s}^{(n)}_{j}-ng_j})/{\sqrt{n}}$ , for all $j\in\{1,\ldots,\kappa-1\}$ . The following convergence in distribution holds in $\mathbb{R}^{2\kappa-1}$ :

\[\big(\overline{{\boldsymbol\ell}}_1^{(n)},\ldots,\overline{{\boldsymbol\ell}}_\kappa^{(n)},\mathbf{x}_1^{(n)},\ldots,\mathbf{x}_{\kappa-1}^{(n)}\big)\xrightarrow[n]{(d)}(\overline{{\boldsymbol\ell}}_1,\ldots,\overline{{\boldsymbol\ell}}_\kappa,\mathbf{x}_1,\ldots,\mathbf{x}_{\kappa-1}),\]

where the variables $\overline{{\boldsymbol\ell}}[\kappa]$ are independent of the $\mathbf{x}[\kappa-1]$ , the r.v. $\overline{{\boldsymbol\ell}}_j$ is exponentially distributed with rate $m_j$ (see equation (7)), and $\mathbf{x}=\mathbf{x}[\kappa-1]$ is a centred Gaussian random vector whose inverse covariance matrix $\Sigma_K^{-1}$ and $\mathbb{d}_K=\det(\Sigma_K^{-1})$ were given in (7). Note that the joint density $\chi_K$ of $(\overline{{\boldsymbol\ell}}_1,\ldots,\overline{{\boldsymbol\ell}}_\kappa,\mathbf{x}_1,\ldots,\mathbf{x}_{\kappa-1})$ is

(33) \begin{align} \chi_K(\ell[\kappa],x[\kappa-1])\,:\!=\prod_{j=1}^{\kappa}m_j\exp\left(-m_j\overline{\ell}_j\right)\cdot\sqrt{\frac{\mathbb{d}_K}{(2\pi)^{\kappa-1}}}\exp\left(-\frac{1}{2} {}^t x\Sigma_K^{-1}x\right). \end{align}

Remark 2.1. Note that we do not use the variable $\displaystyle\mathbf{x}_\kappa^{(n)}\,:\!=({\mathbf{s}^{(n)}_\kappa-ng_\kappa})/{\sqrt{n}}$ , since its data are contained within $\mathbf{x}^{(n)}[\kappa-1]$ as $\mathbf{x}_\kappa^{(n)}=-\sum_{j=1}^{\kappa-1}\mathbf{x}_j^{(n)}$ .

In fact, this theorem states that, asymptotically, $\mathbf{x}^{(n)}[\kappa-1]$ has a density with respect to the Lebesgue measure on $\mathbb{R}^{\kappa-1}.$ But we can also deduce that $(\overline{{\boldsymbol\ell}}_1^{(n)},\ldots,\overline{{\boldsymbol\ell}}_\kappa^{(n)},\mathbf{x}_1^{(n)},\ldots,\mathbf{x}_{\kappa}^{(n)})$ converges in distribution and that, for any bounded continuous test function $\psi\;:\;\mathbb{R}_{+}^\kappa\times\mathbb{R}^{\kappa}\to\mathbb{R}$ ,

$$\mathrm{E}[\psi(\overline{{\boldsymbol\ell}}^{(n)}[\kappa],\mathbf{x}^{(n)}[\kappa])]\underset{n\to\infty}{\longrightarrow} \mathrm{E}[\psi(\overline{{\boldsymbol\ell}}[\kappa],\mathbf{x}_1,\ldots,\mathbf{x}_{\kappa-1},-\mathbf{x}_1-\ldots-\mathbf{x}_{\kappa-1})].$$

Before we get to the proof of this theorem, we recall the following (simple but) powerful lemma that was also used in [Reference Morin14, Lemma 4.3] (and which is proved therein).

Lemma 2.3. Let $(\chi_n)_{n\in\mathbb{N}}$ be a sequence of non-negative measurable functions on $\mathbb{R}^d$ . Assume that, for all $\varepsilon>0$ , there exists a compact set $K_\varepsilon$ such that, for all n large enough, $\int_{K_\varepsilon^c}\chi_n<\varepsilon$ (where $K_\varepsilon^c$ is the complement of $K_\varepsilon$ in $\mathbb{R}^d$ ) and that $\chi_n$ uniformly converges on all compact sets of $\mathbb{R}^d$ towards a density $\chi$ (with respect to the Lebesgue measure on $\mathbb{R}^d$ ). Then, there exists a sequence $(\alpha_n)_{n\in\mathbb{N}}$ such that, for n large enough (for small values of n, $\chi_n$ could be zero), $\frac{1}{\alpha_n}\chi_n$ is a density and $\alpha_n\underset{n\to+\infty}{\longrightarrow}1.$

Proof of Theorem 2.3. The successive steps of this proof are generalizations of those of [Reference Morin14, Theorem 4.1]. The proof should be carried out in two steps, but only the first one is detailed, as the second is pretty much identical to that given in [Reference Morin14, Theorem 4.1]. These steps are as follows.

1. (i) We show the uniform convergence on compact sets of the ‘density’ $f^{(n)}_K$ of the pair $(\overline{{\boldsymbol\ell}}^{(n)}[\kappa],\mathbf{x}^{(n)}[\kappa-1] )$ . More exactly, we show the uniform convergence on compact sets of a density $\chi^{(n)}_K$ , introduced in (37), associated with these random variables to $\chi_K$ .

2. (ii) At this point, we should give an argument of uniform integrability for this limit to apply Lemma 2.3 and conclude. This part will be skipped as it is only a rewriting of the second step of [Reference Morin14, Theorem 4.1].

We detail Step 1. Let $\psi\;:\;\mathbb{R}_{+}^\kappa\times\mathbb{R}^{\kappa-1}\to\mathbb{R}$ be a bounded continuous test function and let us pass to the limit in the expectation

(34) \begin{equation} \mathrm{E}[\psi(\overline{{\boldsymbol\ell}}^{(n)}[\kappa],\mathbf{x}^{(n)}[\kappa-1])] =\sum_{s[\kappa]\in\mathbb{N}_K(n)}\int_{\mathbb{R}^\kappa}\psi\left(n\ell[\kappa],\frac{s-ng}{\sqrt{n}}[\kappa-1]\right)f^{(n)}_{K}(\ell[\kappa],s[\kappa])\mathrm{d}\ell[\kappa], \end{equation}

where the joint distribution $f^{(n)}_{K}$ of the couple $({\boldsymbol\ell}^{(n)}[\kappa],\mathbf{s}^{(n)}[\kappa])$ is given in Theorem 2.2.

We want to perform substitutions of the type $\overline{\ell}^{(n)}_j=n\ell_j$ and $\displaystyle x^{(n)}_j={(s_j-ng_j)}{/\sqrt{n}}$ in the rhs of (34), to prove that $\overline{{\boldsymbol\ell}}^{(n)}_j$ converges to an exponentially distributed variable, and $\mathbf{x}^{(n)}[\kappa-1]$ to a Gaussian vector.

To make the support of the Gaussian vector apparent, let us turn our sum over $s[\kappa]\in\mathbb{N}_K(n)$ into an integral. We set $s_j= \lfloor q_j \rfloor$ for all $j\in\{1\ldots,\kappa\}$ , where $q[\kappa-1]$ browses the set $\overline{\mathbb{N}_{\kappa-1}}(n)\,:\!= \{q[\kappa-1],\text{ with }q_j>0,\text{ such that }\sum_{j=1}^{\kappa-1} \lfloor q_j \rfloor\leq n \}$ , with the convention set to $\left\lfloor q_\kappa\right\rfloor\,:\!=n-\sum_{j=1}^{\kappa-1} \lfloor q_j \rfloor$ (notice that there is no integration with respect to $q_\kappa$ ).

With (31), we get to

\begin{multline*} \mathrm{E}\left[\psi\left(\overline{{\boldsymbol\ell}}^{(n)}[\kappa],\mathbf{x}^{(n)}[\kappa-1]\right)\right] \\ =\mathsf{Faktor}(n)\int_{\overline{\mathbb{N}_{\kappa-1}}(n)}\int_{\mathcal{L}_K}\psi\left(n\ell[\kappa],\frac{\left\lfloor q\right\rfloor-ng}{\sqrt{n}}[\kappa-1]\right)\mathbb{M}(\lfloor q[\kappa]\rfloor)\\ \times\prod_{j=1}^{\kappa} \left(1-\frac{\mathfrak{cl}_j(\ell[\kappa])}{r_j}\right)^{\lfloor q_j\rfloor+\lfloor q_{{j+1}}\rfloor-1}\cdot\mathrm{d}\ell[\kappa]\mathrm{d}q[\kappa-1], \end{multline*}

but let us rather consider the expectation

(35) \begin{multline} \mathrm{E}\left[\psi\left(\overline{{\boldsymbol\ell}}^{(n)}[\kappa],\overline{\mathbf{x}}^{(n)}[\kappa-1]\right)\right]=\mathsf{Faktor}(n)\int_{\overline{\mathbb{N}_{\kappa-1}}(n)}\int_{\mathcal{L}_K}\psi\left(n\ell[\kappa],\frac{{q}-ng}{\sqrt{n}}[\kappa-1]\right)\mathbb{M}(\lfloor q[\kappa]\rfloor)\\ \times\prod_{j=1}^{\kappa} \left(1-\frac{\mathfrak{cl}_j(\ell[\kappa])}{r_j}\right)^{\lfloor q_j\rfloor+\lfloor q_{{j+1}}\rfloor-1}\cdot\mathrm{d}\ell[\kappa]\mathrm{d}q[\kappa-1], \end{multline}

(we withdrew the floor function in $\psi$ ), where for all $j\in\{1,\ldots,\kappa-1\}$ , we set $\overline{\mathbf{x}}^{(n)}_j\,:\!=\mathbf{x}^{(n)}_j+\mathbf{U}_j/\sqrt{n}$ , with $\mathbf{U}_j$ a r.v. uniformly distributed in [0, 1].

We replaced a sum by an integral and this amounts to representing a discrete random variable as a continuous one, i.e. if X has a discrete law, $\mathrm{P}(X=k)=p_k,k\in\mathbb{Z}$ , then $X\mathrel{\mathop{\kern 0pt=}\limits^{(d)}}\lfloor X+U\rfloor$ , where U is uniform in [0, 1]. Then,

\[\sum_{k\in\mathbb{Z}} p_kf(k)=\int_\mathbb{R} f \left (\lfloor x\rfloor \right ) p_{\lfloor x\rfloor}\mathrm{d}x.\]

We are going to prove first that $\mathrm{E} [\psi (\overline{{\boldsymbol\ell}}^{(n)}[\kappa],\overline{\mathbf{x}}^{(n)}[\kappa-1] ) ]$ converges to deduce that its counterpart $\mathrm{E} [\psi (\overline{{\boldsymbol\ell}}^{(n)}[\kappa],{\mathbf{x}}^{(n)}[\kappa-1] ) ]$ converges as well, to the same limit.

We can finally make our substitutions $\overline{\ell}_j\,:\!=\overline{\ell}^{(n)}_j=n\ell_j$ , for all $j\in\{1,\ldots,\kappa\}$ , and $\displaystyle x_j\,:\!=x^{(n)}_j={(q_j-ng_j)}/{\sqrt{n}}$ , for all $j\in\{1,\ldots,\kappa-1\}$ , in (35), where we have hidden the dependence on n to unburden our equations. We reach

(36) \begin{multline} \mathrm{E}\left[\psi\left(\overline{{\boldsymbol\ell}}^{(n)}[\kappa],\overline{\mathbf{x}}^{(n)}[\kappa-1]\right)\right] \\ =\int_{\mathbb{R}^{\kappa-1}}\int_{\mathbb{R}_+^\kappa}\psi \left (\overline{\ell} [\kappa],x[\kappa-1] \right )\chi^{(n)}_K \left (\overline{\ell}[\kappa],x[\kappa-1] \right )\mathrm{d}\overline{\ell}[\kappa]\mathrm{d}x[\kappa-1], \end{multline}

where, for all $n\geq 3,$ $\chi^{(n)}_K$ stands for the joint distribution of the pair $(\overline{{\boldsymbol\ell}}^{(n)}[\kappa],\overline{\mathbf{x}}^{(n)}[\kappa-1])$ . The function $\chi^{(n)}_K$ can be decomposed as follows:

(37) \begin{align} \chi^{(n)}_K\left(\overline{\ell}[\kappa],x[\kappa-1]\right)\,:\!=\omega(n,K)\ h^{(1)}_n \left (\overline{\ell}[\kappa],x[\kappa-1] \right)\ h^{(2)}_n \left (x[\kappa-1] \right), \end{align}

with

(38) \begin{multline} \omega(n,K)=\Bigg[\prod_{j=1}^\kappa\frac{f_j+f_{{j-1}}}{\sin\!(\theta_j)r_j}\cdot\frac{n!\ \mathrm{e}^{3n}(\bar{f})^{3n-\kappa}}{\widetilde{\mathrm{P}}_K(n)\ n^{3n-\kappa}}\Bigg]\Bigg[\frac{1}{n^\kappa\sqrt{(2\pi)^{\kappa+1}\mathbb{d}_K}\prod_{j=1}^\kappa\sqrt{g_j(g_j+g_{{j+1}})}}\Bigg]\\\times\Bigg[\prod_{j=1}^\kappa\frac1{m_j}\Bigg]\frac{1}{n^{(\kappa+1)/2}}, \end{multline}

which can be rewritten as

(39) \begin{align} \omega(n,K)&=\frac{1}{\sqrt{(2\pi)^{\kappa+1}\mathbb{d}_K}}\cdot\Bigg[\prod_{j=1}^\kappa\sqrt{\frac{f_j+f_{{j-1}}}{f_j}}\frac{1}{m_j\sin\!(\theta_j)r_j}\Bigg]\cdot\frac{n!\ \mathrm{e}^{3n}(\bar{f})^{3n}}{\widetilde{\mathrm{P}}_K(n)\ n^{3n}}\frac{1}{n^{(\kappa+1)/2}}, \end{align}

and

(40) $${h^{(1)}_n\left(\overline{\ell}[\kappa],x[\kappa-1]\right) =\mathbb{1}_{\overline{\ell}[\kappa] \in n\mathcal{L}_K}\ \prod_{j=1}^\kappa m_j\left(1-\frac{\mathfrak{cl}_j(\ell[\kappa])}{n\cdot r_j}\right)^{d^{(2)}_j(x[\kappa-1])},}$$

(41) $${h^{(2)}_n\left(x[\kappa-1]\right) =\sqrt{\frac{\mathbb{d}_K}{(2\pi)^{\kappa-1}}}\prod_{j=1}^\kappa (2\pi n)\sqrt{g_j(g_j+g_{{j+1}})}\cdot\mathbb{M}(d^{(1)}(x[\kappa-1])),}$$

where, with the convention $x_\kappa\,:\!=-\sum_{j=1}^{\kappa-1}x_j$ , we set

(42) $${d^{(1)}_j\left(x[\kappa-1]\right) =\lfloor ng_j + \sqrt{n}x_j \rfloor, \quad \text{ for all }j\in\{1,\ldots,\kappa\},}$$

(43) $${ d^{(2)}_j\left(x[\kappa-1]\right) =\lfloor ng_j + \sqrt{n}x_j \rfloor+\lfloor ng_{j-1} + \sqrt{n}x_{j-1} \rfloor-1, \quad \text{ for all }j\in\{1,\ldots,\kappa\}. }$$

Notice here that there is a small approximation with this convention: indeed, with this change of variable, the natural definition of $d^{(1)}_\kappa\left(x[\kappa-1]\right)$ in the integral corresponding to $\mathrm{E}[\psi (\overline{{\boldsymbol\ell}}^{(n)}[\kappa],\overline{\mathbf{x}}^{(n)}[\kappa-1] ) ]$ should be $d^{(1)}_\kappa (x[\kappa-1] )\,:\!=n-{\sum_{j=1}^{\kappa-1} \lfloor\sqrt{n}x_j+ng_j \rfloor}$ , to be consistent (the same problem appears for $d^{(2)}_{j}= \lfloor q_j \rfloor+ \lfloor q_{j+1} \rfloor-1$ when $j\in\{\kappa-1,\kappa\})$ . This means that the definitions (42) and (43) correspond to the expectation $\mathrm{E} [\psi (\overline{{\boldsymbol\ell}}^{(n)}[\kappa],\widetilde{\mathbf{x}}^{(n)}[\kappa-1] ) ]$ for some variables $\widetilde{\mathbf{x}}^{(n)}[\kappa-1]$ close to $\overline{\mathbf{x}}^{(n)}[\kappa-1]$ . But it is clear that, with

\[ \frac{n-{\sum_{j=1}^{\kappa-1}\left\lfloor\sqrt{n}x_j+ng_j\right\rfloor}}{\left\lfloor\sqrt{n}x_\kappa+ng_\kappa\right\rfloor}\underset{n\to\infty}{\longrightarrow} 1, \]

we can easily prove that

\[ \vert\overline{\mathbf{x}}^{(n)}[\kappa-1]-\widetilde{\mathbf{x}}^{(n)}[\kappa-1] \vert\xrightarrow[n]{\mathrm{(proba.)}} 0 \]

and thus, by Slutsky’s lemma, the limits in distribution of $(\overline{{\boldsymbol\ell}}^{(n)}[\kappa],\overline{\mathbf{x}}^{(n)}[\kappa-1] )$ and $ (\overline{{\boldsymbol\ell}}^{(n)}[\kappa],\widetilde{\mathbf{x}}^{(n)}[\kappa-1] )$ are the same.

We have arranged the factors so that, as we will see, $h^{(1)}_n$ and $h^{(2)}_n$ converge to some probability densities (and since a density has integral 1, this will be a crucial point for the conclusion).

Note first that there exists $\eta>0$ such that $[0,\eta]^\kappa\subset \mathcal{L}_K$ ; thus, we have $n\mathcal{L}_\kappa\underset{n\to\infty}{\longrightarrow} \mathbb{R}_{+}^\kappa$ . Then for every compact $H\subset \mathbb{R}_{+}^\kappa$ , and for all $\varepsilon>0,$ there exists $n_0\in\mathbb{N}$ such that, for all $n\geq n_0,$ $H\subset n\mathcal{L}_K$ , i.e. $ \Vert\mathbb{1}_{\{{\overline{\ell}[\kappa]\in n\mathcal{L}_K}\}}-1\Vert=0<\varepsilon $ , so that the map $ \overline{\ell}[\kappa]\mapsto\mathbb{1}_{\{{\overline{\ell}[\kappa]\in n\mathcal{L}_K}\}} $ converges uniformly to the constant function 1 on every compact set of $\mathbb{R}_+^\kappa$ . Now by the standard approximation

\[ \left(1-\frac{a}{n}\right)^{nb}\underset{n\to+\infty}{\longrightarrow}\mathrm{e}^{-ab}, \]

uniformly for (a, b) on every compact set of $\mathbb{R}^2$ , and using additionally the fact that $\mathbb{1}_{\overline{\ell}[\kappa]\in n\mathcal{L}_K}\underset{n\to+\infty}{\longrightarrow}1$ uniformly on every compact set of $\mathbb{R}_+^\kappa$ , we get that $h^{(1)}_n$ converges uniformly on every compact set of $\mathbb{R}_{+}^\kappa\times\mathbb{R}^{\kappa-1}$ towards $h^{(1)}$ , with

(44) \begin{align} h^{(1)}(\overline{\ell}[\kappa],x[\kappa-1])=\prod_{j=1}^{\kappa}m_j\exp\left(-m_j\overline{\ell}_j\right). \end{align}

Now, thanks to the local limit theorem for Poisson r.v. [Reference Petrov15, Theorem VII.1.1], $h_n^{(2)}$ converges uniformly towards $h^{(2)}$ , where $h^{(2)}$ is defined for $x[\kappa-1]\in\mathbb{R}^{\kappa-1},$ as

(45) \begin{align} h^{(2)}(x[\kappa-1])\,:\!=\sqrt{\frac{\mathbb{d}_K}{(2\pi)^{\kappa-1}}}\prod_{j=1}^\kappa\exp\left(-\frac{1}{2} \left[\frac{x_j^2}{g_j}+\frac{(x_j+x_{{j+1}})^2}{g_j+g_{{j+1}}}\right]\right). \end{align}

The map $h^{(2)}$ can be rewritten as

\[h^{(2)}(x[\kappa-1])=\sqrt{\frac{\mathbb{d}_K}{(2\pi)^{\kappa-1}}}\exp\left(-\frac{1}{2} {}^t x\Sigma_K^{-1}x\right),\]

where $\Sigma_K^{-1}$ , given in Theorem 1.3, corresponds to the matrix associated to the exponential product in (45), and $\mathbb{d}_K=\det\left(\Sigma_K^{-1}\right)$ is its determinant.

We have established the following uniform convergence on every compact set of $\mathbb{R}_+^{\kappa}\times\mathbb{R}^{\kappa-1}$ :

(46) \begin{align} \frac{1}{\omega(n,K)}\chi^{(n)}_K \left (\overline{\ell}[\kappa],x[\kappa-1] \right )\underset{n\to +\infty}{\longrightarrow} \chi_{K} \left (\overline{\ell}[\kappa],x[\kappa-1] \right ). \end{align}

This concludes the first step of our proof.

As said in the introduction, the uniform integrability is the same as in [Reference Morin14, Theorem 4.1], so we skip this step; the reader is directed to [Reference Morin14] for the remaining details. However, the conclusion of this proof is the same.

This proves that $\chi^{(n)}_{K}$ converges pointwise to $\chi_{K}$ or, by definition, that

\[ (\overline{{\boldsymbol\ell}}^{(n)}[\kappa],\overline{\mathbf{x}}^{(n)}[\kappa-1] )\xrightarrow[n]{(d)} (\overline{{\boldsymbol\ell}}[\kappa],\mathbf{x}[\kappa-1] ). \]

Now, since

\[ \vert\overline{\mathbf{x}}^{(n)}[\kappa-1]-\mathbf{x}^{(n)}[\kappa-1]\vert\xrightarrow[n]{(\mathrm{proba.})}0, \]

then, by Slutsky’s lemma,

\[(\overline{{\boldsymbol\ell}}^{(n)}[\kappa],{\mathbf{x}}^{(n)}[\kappa-1] )\xrightarrow[n]{(d)} (\overline{{\boldsymbol\ell}}[\kappa],\mathbf{x}[\kappa-1] ).\]

At the same time, we find that $\omega(n,K)\underset{n\to+\infty}{\longrightarrow}1$ , which, with Stirling’s formula, leads to

(47) \begin{align} \widetilde{\mathrm{P}}_K(n)\underset{n\to+\infty}{\sim} \frac{1}{(2\pi)^{\kappa/2}\sqrt{\mathbb{d}_K}}\cdot\Bigg[\prod_{j=1}^\kappa\sqrt{\frac{f_j+f_{{j-1}}}{f_j}}\frac{1}{m_j\sin\!(\theta_j)r_j}\Bigg]\cdot\frac{\mathrm{e}^{2n}(\bar{f})^{3n}}{n^{2n+\kappa/2}}. \end{align}

Theorem 1.2 shows, in particular, that $ (\bar{f})^{3n}={\mathrm{AP}^*(K)^{3n}} /{4^n}$ , and that ${f_j} / ({f_j+f_{{j-1}}})=w_j$ .

3.1. The general result: proofs of Lemma 1.1 and Theorem 1.4

The idea of these proofs is to find a sort of ‘converse’ to the previous lemma. Indeed, Lemma 3.2 allows one to shrink a convex polygon while keeping its limit shape intact, as long as the limit shape is not shrunk at the same time. Is there any way we could ‘increase’ (in some sense) a convex polygon, while keeping its limit shape intact?

Fix $\kappa\geq3$ and let us rather assume that K is in $\mathbf{P}_{\kappa+1}$ , and that $\mathsf{Dom}(K)$ is tangent to exactly $\kappa$ sides of K (that is, all sides except one). We focus on this case for now, and explain at the end why the same arguments as those developed in the following stay valid for any K in $\mathbf{P}_{\kappa+j}$ , $j\geq1$ , having $\mathsf{Dom}(K)$ tangent to exactly $\kappa$ sides of K.

Denote by $\vec{n}_1,\ldots,\vec{n}_{\kappa+1}$ the successive sides of K, viewed as vectors (or directions) in this proof. By hypothesis, the set $\mathsf{Dom}(K)$ is tangent to K on $\kappa$ sides; thus, we assume that the $(\kappa+1)$ th side is the missing one, as Figure 9. Consider the $\kappa$ straight lines directed by $\left(\vec{n}_i\right)_{1\leq i\leq\kappa}$ ; they delineate a convex polygon $K_{\mathcal{T}}\in\mathbf{P}_\kappa$ (with $\kappa$ sides) that contains K. We let $\mathsf{X}$ be the vertex of $K_{\mathcal{T}}$ , being the intersection point between the lines of directions $\vec{n}_\kappa$ and $\vec{n}_1.$

Figure 9.

An example of polygon K whose limit shape (the set with smooth boundary contained inside K) is not tangent to one side (the one that is not dotted). Consider the larger polygon, which has no side with direction $\vec{n}_{\kappa+1}$ , and has vertex $\mathsf{X}$ instead: this polygon is the polygon $K_{\mathcal{T}}$ : the boundary of $\mathsf{Dom}(K)$ is now tangent to every side of $K_{\mathcal{T}}$ . We will show that $\mathsf{Dom}(K)=\mathsf{Dom}(K_{\mathcal{T}})$ .

Let us prove that $K_{\mathcal{T}}$ satisfies (10), (11), and (12). To do this, consider the family of convex polygons $(K_{t})_{0\leq t\leq1}$ defined as follows:

1. (i) $K_0=K$ .

2. (ii) $K_1=K_{\mathcal{T}}$ .

3. (iii) For $t>0$ , $K_t$ is the polygon $K_0$ where we ‘pushed back’ at distance t the $(\kappa+1)$ th side, with direction $\vec{n}_{\kappa+1}$ (which is thus at distance $1-t$ from $\mathsf{X}$ ). More precisely, this is the intersection between $K_{\mathcal{T}}$ and the halfplane delineated by the straight line with direction $\vec{n}_{\kappa+1}$ at distance $1-t$ from X. Of course, we have, for all $t\in[0,\,1]$ , $K_t\subset K_{\mathcal{T}}$ .

This concept is illustrated in Figure 10. Note that, with this definition of $(K_t)$ , we have, for all $t\in[0,\,1]$ ,

(49) \begin{align} d_{\mathrm{H}}(K_s,K_t)\underset{s\to t}{\longrightarrow} 0. \end{align}

Figure 10.

Polygon $K_t$ , drawn at times $t=0$ and $t=1$ , and its limit shape, which is the set with smooth boundary contained within $K_t$ . We will prove that these limit shapes are indeed the same set.

We are going to prove that, for all $t\in[0,\,1],$ we have $\mathsf{Dom}(K_t)=\mathsf{Dom}(K)$ . To do this, we prove that the map $F\;:\;t\mapsto \mathrm{AP}^*(K_t)$ is continuous, and then we will deduce that it must be constant on [0, 1].

1. The map $F\;:\;t\mapsto \mathrm{AP}^*(K_t)$ is continuous. First, notice that, since the affine perimeter is an upper-semicontinuous map (a proof of this claim can be found in [Reference Lutwak12, Proposition 4.4]), so is F. In addition, F is non-decreasing on [0, 1], by definition.

Let us assume now that F is not continuous, and denote by $t^*>0$ the moment of a jump of F, so that $\limsup_{t\to t*}F(t)<F(t^*)$ . Then let $\alpha$ be a real value in the open interval $(\limsup_{t\to t*}F(t),F(t^*))$ .

There exists a dilatation by some $0<\lambda<1$ , such that $\mathrm{AP}(\lambda\mathsf{Dom}(K_{t^*}))=\lambda^{2/3}F(t^*)=\alpha$ . Since $\lambda\mathsf{Dom}(K_{t^*})$ is just a ‘shrunk’ version of $\mathsf{Dom}(K_{t^*})$ , there exists $0<\tilde{t}<t^*$ with $\lambda\mathsf{Dom}(K_{t^*})\subset K_{\tilde{t}}$ . This means, in particular, that $F(\tilde{t})\geq \alpha>\limsup_{t\to t*}F(t)$ and then $F(\tilde{t})=F(t^*)$ because F has no image in the interval $(\limsup_{t\to t*}F(t),F(t^*))$ . Since F is non-decreasing, F is constant on the interval $[\tilde{t},t^*]$ ; this is absurd, since $\limsup_{t\to t*}F(t)<F(t^*)$ .

2. The map $F\;:\;t\mapsto \mathrm{AP}^*(K_t)$ is constant. Assume it is not: there exists $t>0$ such that we have $F(t)>F(0)$ . By continuity, there exists $s\geq0$ such that $[s,t]\subseteq [0,t]$ and $F([s,t])=[F(0),F(t)]$ , and F is constant on [0,s].

For n large enough, there exists a sequence $(s_n)\in [s,t]$ with $F(s_n)=F(s)+1/n$ . Consider the sequence $(\mathsf{D}_{s_n})_n\,:\!=(\mathsf{Dom}(K_{s_n}))_n$ ; it is a sequence of compact convex sets that are all subsets of $K_1$ and, since the set of compact convex subsets of $K_1$ is compact for $d_{\mathrm{H}}$ [Reference IstrăŢescu10], there exists a subsequence $n_k$ such that $(\mathsf{D}_{s_{n_k}})$ converges to a convex set $\bar{\mathsf{D}}$ for the Hausdorff distance.

By upper-semicontinuity of the affine perimeter, we have $\mathrm{AP}(\bar{\mathsf{D}})\geq \limsup_{k\to+\infty} F(s_{n_k})=F(s).$ In addition, since $\bar{\mathsf{D}}=\lim_{k\to+\infty} \mathsf{D}_{s_{n_k}} $ and $\mathsf{D}_{s_{n_k}}\subseteq K_{s_{n_k}}$ , we have $\bar{\mathsf{D}}\subseteq K_s$ , since $d_{\mathrm{H}}(K_{s_{n_k}},K_s)\underset{k\to+\infty}{\longrightarrow}0$ by (49). By unicity, this means $\bar{\mathsf{D}}=\mathsf{Dom}(K_s)=\mathsf{Dom}(K_0).$

Now let $\mathscr{L}_s$ be the segment corresponding to the $(\kappa+1)$ th side of $K_s$ , the one we added with direction $\vec{n}_{\kappa+1}$ at distance $(1-s)$ of $\mathsf{X}$ .

All sets $\mathsf{D}_{s_n}$ (for n large enough) intersect with $K_s$ (because $F(s_n)>F(s)$ ; thus, $\mathsf{D}_{s_n}\subsetneq K_s$ would go against the fact that $\mathsf{D}_s$ realizes the maximum of the affine perimeter among convex subsets of $K_s$ ), meaning that we have $d_{\mathrm{H}}(\mathsf{D}_{s_{n_k}},\mathscr{L}_s)=0$ ; hence, when $k\to+\infty$ , we get $d_{\mathrm{H}}(\bar{\mathsf{D}},\mathscr{L}_s)=0$ , by continuity. But since $\mathsf{Dom}(K_0)=\mathsf{Dom}(K_s)=\bar{\mathsf{D}}\subset K_s$ , this means that $\mathsf{Dom}(K_0)$ must intersect (or at least be tangent with) $\mathscr{L}_s$ , which is impossible because $d_{\mathrm{H}}(\mathsf{Dom}(K_0),\mathscr{L}_s)\geq d_{\mathrm{H}}(\mathsf{Dom}(K_0),\mathscr{L}_0)$ , and we assumed that $d_{\mathrm{H}}(\mathsf{Dom}(K_0),\mathscr{L}_0)>0$ .

This proves that F is constant; hence, $\mathsf{Dom}(K_{1})=\mathsf{Dom}(K_{0})=\mathsf{Dom}(K_{\mathcal{T}})$ and this proves (10) and (11) simultaneously.

To prove (12), the properties $\mathsf{Dom}(K)=\mathsf{Dom}(K_{\mathcal{T}})$ and $K\subset K_{\mathcal{T}}$ , yield that, for $\varepsilon>0$ small enough, there exists $N\in\mathbb{N}$ such that, for all $n\geq N$ , we have $S_n(K,\varepsilon)=S_n(K_{\mathcal{T}},\varepsilon)$ (see (18)). The last conclusion stated in Theorem 2.1 enables this conclusion.

3. A remark about how we may replace $\mathscr{L}_s$ . In the second point of this proof, we use the fact that $d_{\mathrm{H}}(\mathsf{D}_{s_{n_k}},\mathscr{L}_s)=0$ and we take the limit on k to obtain $d_{\mathrm{H}}(\bar{\mathsf{D}},\mathscr{L}_s)=0$ . In this argument, we do not use the fact that $\mathscr{L}_s$ is a segment. Assume now that, instead of the $(\kappa+1)$ th side of $K_0$ , i.e. the segment $\mathscr{L}_0$ , we have a concave closed shape $\mathscr{S}_0$ joining the extremities of the 1st and $\kappa$ th sides of $K_0$ and forming a convex set $S_0$ , as long as $S_0\subseteq K_1$ . We still have $\mathsf{Dom}(K_0)\subset S_0$ and $d_{\mathrm{H}}(\mathsf{Dom}(K_0),\mathscr{S}_0)>0$ . Two examples of sets $S_0$ are given in Figure 11.

Figure 11.

Two kinds of convex body $S_0$ , drawn at times $t=0$ , with the same smooth limit shape contained inside, which is $\mathsf{Dom}(K_0)$ . On the left, the concave shape $\mathscr{S}_0$ is a smooth map; on the right, $\mathscr{S}_0$ is composed of several polygonal lines.

Let us further define a family of increasing convex sets $(S_t)_{t\in[0,\, 1]}$ such that the following hold:

1. (i) $S_0$ is the set we just defined;

2. (ii) the function $t\mapsto S_t$ is non-decreasing for the inclusion order, continuous for $d_{\mathrm{H}}$ and satisfies $S_1 = K_1$ .

Under this condition, this means that, for all $t\in[0,\,1)$ , the boundary of the set $S_t$ is composed of the $\kappa$ polygonal lines forming the closed set $\partial K_1\cap\partial S_t $ , which are joined together by a concave closed shape $\mathscr{S}_t$ lying in the interior of $K_1$ .

Had we used this family instead of $(K_t)$ in the proof of point 2, this would have just changed the closed domain $\mathscr{L}_s$ into $\mathscr{S}_s$ and the rest of the proof would have remained the same. In the end, we would get a slightly better result, as this leads to the fact that $\mathsf{Dom}(K_0)=\mathsf{Dom}(S_t)=\mathsf{Dom}(K_1)$ for all $t\in[0,\,1]$ .

4. The case $\mathbf{P}_{\kappa+j}$ . We can now generalize this reasoning to the case where K is in $\mathbf{P}_{\kappa+j}$ , $j\geq2$ , and $\mathsf{Dom}(K)$ is tangent to exactly $\kappa$ sides of K. Assume first that the j sides to which $\mathsf{Dom}(K)$ is not tangent are consecutive. Denote once more by $\vec{n}_1,\ldots,\vec{n}_{\kappa+j}$ the successive sides of K, and assume first that the last j sides are the ‘missing’ ones. This means that there is an arc of parabola joining the $n_\kappa$ th and $n_1$ th sides, to which it is also tangent. In particular, the angle between the $n_\kappa$ th and $n_1$ th side must be closing, or the parabola could not be tangent to these sides. Consider the $\kappa$ straight lines directed by $\left(\vec{n}_i\right)_{1\leq i\leq\kappa}$ ; they delineate a convex polygon $K_{\mathcal{T}}\in\mathbf{P}_\kappa$ (with $\kappa$ sides) that contains K.

This means that we can construct a family $(S_t)$ satisfying conditions (i) and (ii) of point 3. (just as on the right of Figure 11), where the concave closed shape $\mathscr{S}_0$ comprises the last j ‘missing’ sides lying in the interior of $S_1\,:\!=K_{\mathcal{T}}$ . In this case, we have $S_0\,:\!=K$ ; by the previous point we obtain $\mathsf{Dom}(K_{\mathcal{T}})=\mathsf{Dom}(K)$ , which proves this case of consecutive ‘missing’ sides.

If the j sides to which $\mathsf{Dom}(K)$ is not tangent are not consecutive, we can still denote by $\vec{n}_1,\ldots,\vec{n}_{\kappa+j}$ the directions of the successive sides of K, and assume that the last i sides, with $1\leq i<j$ , are ‘missing’ sides. Consider the $\kappa+j-i$ straight lines directed by $\left(\vec{n}_q\right)_{1\leq q\leq\kappa+j-i}$ ; they delineate a convex polygon $K_1$ with $\kappa+j-i$ sides that contains K. The exact same arguments as before allow us to prove that $K\in\mathcal{S}(K_1)$ and

\[\mathsf{Area}(K)^n\ \mathrm{P}_{K}(n)\underset{n\to\infty}{\sim} \mathsf{Area}(K_1)^n\ \mathrm{P}_{K_1}(n).\]

Step by step, treating each bunch of consecutive missing sides one after the other, there will exist a finite sequence of convex polygons $K_1\subset\ldots\subset K_k= K_{\mathcal{T}}$ , with $K_{\mathcal{T}}\in\mathbf{P}_{\kappa}^{\mathcal{T}}$ (with $\kappa$ sides) and $K_p\in\mathcal{S}(K_{p+1})$ , such that

\[\mathsf{Area}(K_p)^n\ \mathrm{P}_{K_p}(n)\underset{n\to\infty}{\sim} \mathsf{ Area}(K_{p+1})^n\ \mathrm{P}_{K_{p+1}}(n)\]

for all $p\in\{1,\ldots,k-1\}$ , so that the last step amounts to proving the previous case. This proves (12).

4.1. Algorithm

Set $\mathrm{AP}^*=0$ and $I^*=\emptyset$ . These quantities will keep the current best candidate for $K_{\mathcal{T}}$ in the memory.

For all $I\in \mathsf{Valid}(K)$ do:

Step 1: Consider the solution to the system in the unknowns $f^{(I)}[I]$ :

\[\left\{ \begin{array}{rll}\displaystyle f^{(I)}_j \big (\,f^{(I)}_j+f^{(I)}_{{j+1}} \big ) \big (\,f^{(I)}_j+f^{(I)}_{{j-1}} \big )& = r^{(I)}_j\cdot r^{(I)}_{{j+1}}\cdot\sin \big (\theta^{(I)}_j \big ),&\qquad j\in I,\\ f^{(I)}_j&\geq 0,&\qquad j\in I. \end{array} \right. \]

As said before, we know that such a solution to this system exists, but it is rarely explicit; a numerical resolution is to be considered. Now, the family $f^{(I)}[I]$ encodes a curve composed of parabola arcs tangent to every side of $K_I$ . We therefore need to determine the tangency points.

Step 2: Compute

\begin{align*} x^{(I)}_j=\frac{f^{(I)}_j}{f^{(I)}_{{j-1}}+f^{(I)}_j},\qquad j\in I. \end{align*}

Step 3: Compute the set $S_I$ defined as the convex set whose boundary is composed of parabola arcs between the points determined by $x^{(I)}[I]$ on the sides of $K_I$ . By construction, we have $S_I=\arg \max_{S\in\mathcal{Z}_{K_I}}\mathrm{AP}(S).$

Step 4: Check that $S_I$ is contained in K. If not, reject this I. If it is, compute $\mathrm{AP}(S_I)$ using (28). If $\mathrm{AP}^*\leq \mathrm{AP}(S_I)$ then we update the current best candidate for $\mathsf{Dom}(K)$ ; thus, we set

\[I^*\leftarrow I \qquad \text{ and } \qquad \mathrm{AP}^*\leftarrow \mathrm{AP}(S_I).\]

4.1.1. Conclusion.

Among all I, one is realizing the convex polygon $K_I=K_{\mathcal{T}}$ of Theorem 1.4. Therefore,

\[\mathrm{AP}^*(K)=\max_{I\in \mathsf{Valid}(K)}\mathrm{AP}(S_I). \]

The final value $I^*$ , which we kept out of Step 3, gives $\mathsf{Dom}(K)=S_{I^*}.$