Discrete & Computational Geometry

, Volume 53, Issue 1, pp 226–244 | Cite as

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

  • Mareen Beermann
  • Matthias ReitznerEmail author


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.


Poisson polytope Random polytope Generating function Efron’s identity 

Mathematics Subject Classification

Primary 60D05 Secondary 60G55 52A22 



M. Beermann was supported in part by the FWF project P 22388-N13, ‘Minkowski valuations and geometric inequalities’. We are grateful to an anonymous referee for careful reading of the manuscript and numerous helpful suggestions.


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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of OsnabrueckOsnabrueckGermany

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