Poisson polytopes

TL;DR

This paper establishes a central limit theorem for the volume and face counts of Poisson random polytopes formed within a fixed convex polytope, advancing understanding of their probabilistic geometric properties.

Contribution

It proves the central limit theorem for volume and $f$-vector of Poisson polytopes in a fixed convex body, a novel result in stochastic geometry.

Findings

Central limit theorem proven for volume of Poisson polytopes

Central limit theorem proven for $f$-vector of Poisson polytopes

Results applicable to convex polytopes in any fixed dimension

Abstract

We prove the central limit theorem for the volume and the $f$-vector of the Poisson random polytope $\Pi_{\eta}$ in a fixed convex polytope $P\subset\mathbb{R}^d$. Here, $\Pi_{\eta}$ is the convex hull of the intersection of a Poisson process $X$ of intensity $\eta$ with $P$.