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

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

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