Beyond the Efron-Buchta identities: distributional results for Poisson polytopes

TL;DR

This paper extends identities related to the distributional properties of Poisson polytopes, providing new moment generating function identities for their measures and vertex counts, generalizing previous results for i.i.d. convex hulls.

Contribution

It introduces new identities involving the moment generating functions of Poisson polytopes, generalizing Efron and Buchta's results for i.i.d. convex hulls.

Findings

Identities for the moment generating function of the measure of Poisson polytopes.

Identities for the number of vertices and non-vertices of Poisson polytopes.

Generalization of existing identities for convex hulls of i.i.d. points.

Abstract

Let $\Pi$ be a random polytope defined as the convex hull of the points of a Poisson point process. Identities involving the moment generating function of the measure of $\Pi$, the number of vertices of $\Pi$ and the number of non-vertices of $\Pi$ are proven. Equivalently, identities for higher moments of the mentioned random variables are given. This generalizes analogous identities for functionals of convex hulls of i.i.d points by Efron and Buchta.