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

Abstract

Let $$\varPi $$? 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 $$\varPi $$?, the number of vertices of $$\varPi $$?, and the number of non-vertices of $$\varPi $$? 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.