Beyond the Efron-Buchta Identities: Distributional Results for Poisson Polytopes

Abstract

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