Variance Asymptotics and Scaling Limits for Random Polytopes

TL;DR

This paper investigates the asymptotic behavior of random polytopes formed by Poisson points in convex sets, providing new scaling limits near vertices and variance asymptotics for geometric functionals, resolving open questions.

Contribution

It establishes scaling limits and variance asymptotics for random polytopes in convex sets, especially near vertices, using germ-grain models, and addresses an open problem in the field.

Findings

Scaling limits for the boundary near vertices of K λ

Variance asymptotics for volume and k-face functionals

Description of limits via cone-like germ-grain models

Abstract

Let K be a convex set in R d and let K $\lambda$ be the convex hull of a homogeneous Poisson point process P $\lambda$ of intensity $\lambda$ on K. When K is a simple polytope, we establish scaling limits as $\lambda$ $\rightarrow$ $\infty$ for the boundary of K $\lambda$ in a vicinity of a vertex of K and we give variance asymptotics for the volume and k-face functional of K $\lambda$, k $\in$ {0, 1, ..., d -- 1}, resolving an open question posed in [18]. The scaling limit of the boundary of K $\lambda$ and the variance asymptotics are described in terms of a germ-grain model consisting of cone-like grains pinned to the extreme points of a Poisson point process on R d--1 $\times$ R having intensity $\sqrt$ de dh dhdv.