The support function of the high-dimensional Poisson polytope

TL;DR

This paper investigates the asymptotic behavior of the support function of high-dimensional Poisson polytopes, identifying regimes based on intensity and deriving distributional limits in each case.

Contribution

It introduces a detailed analysis of the support function's asymptotics for Poisson polytopes in high dimensions, including multiple directions and different intensity regimes.

Findings

Three regimes (subcritical, critical, supercritical) identified based on intensity.

Distributional convergence of the support function established in each regime.

Partial results obtained for the radius-vector function of the polytope.

Abstract

Let $K_\lambda^d$ be the convex hull of the intersection of the homogeneous Poisson point process of intensity $\lambda$ in $\mathbb{R}^d$, $d \ge 2$, with the Euclidean unit ball $\mathbb{B}^d$. In this paper, we study the asymptotic behavior as $d\to\infty$ of the support function $h_\lambda^{(d)}(u) :=\sup_{x\in K_\lambda^d}\langle u,x\rangle$ in an arbitrary direction $u \in {\mathbb S}^{d-1}$ of the Poisson polytope $K_\lambda^d$. We identify three different regimes (subcritical, critical, and supercritical) in terms of the intensity $\lambda:=\lambda(d)$ and derive in each regime the precise distributional convergence of $h_\lambda^{(d)}$ after suitable scaling. We especially treat this question when the support function is considered over multiple directions at once. We finally deduce partial counterparts for the radius-vector function of the polytope.