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

Abstract

Let $Î $ 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 $Î $, the number of vertices of $Î $ and the number of non-vertices of $Î $ 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.