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.
Similar content being viewed by others
References
Affentranger, F.: The expected volume of a random polytope in a ball. J. Microsc. 151, 277–287 (1988)
Bárány, I.: Random polytopes in smooth convex bodies. Mathematika 39, 81–92 (1992)
Bárány, I., Reitzner, M.: Poisson polytopes. Ann. Probab. 38, 1507–1531 (2010)
Bárány, I., Reitzner, M.: On the variance of random polytopes. Adv. Math. 225, 1986–2001 (2010)
Buchta, C.: Stochastische Approximation konvexer Polygone. Z. Wahrsch. Verw. Geb. 67, 283–304 (1984)
Buchta, C.: Zufallspolygone in konvexen Vielecken. J. Reine Angew. Math. 347, 212–220 (1984)
Buchta, C.: An identity relating moments of functionals of convex hulls. Discrete Comput. Geom. 33, 125–142 (2005)
Buchta, C., Müller, J.: Random polytopes in a ball. J. Appl. Probab. 21, 753–762 (1984)
Buchta, C., Reitzner, M.: Equiaffine inner parallel curves of a plane convex body and the convex hulls of randomly chosen points. Probab. Theory Relat. Fields 108, 385–415 (1997)
Buchta, C., Reitzner, M.: The convex hull of random points in a tetrahedron: solution of Blaschke’s problem and more general results. J. Reine Angew. Math. 536, 1–29 (2001)
Calka, P., Yukich, J.E.: Variance asymptotics for random polytopes in smooth convex bodies. Probab. Theory Relat. Fields 152, 435–463 (2014)
Doetsch, G.: Handbuch der Laplace-Transformation II. Birkhäuser, Basel, Stuttgart (1955)
Efron, B.: The convex hull of a random set of points. Biometrika 52, 331–343 (1965)
Hug, D.: Random polytopes. In: Spodarev, E. (ed.) Stochastic Geometry, Spatial Statistics and Random Fields. Lecture Notes in Mathematics, vol. 2068, pp. 205–238, Springer, Heidelberg (2013)
Hug, D., Munsonius, G.O., Reitzner, M.: Asymptotic mean values of Gaussian polytopes. Beitr. Algebra Geom. 45, 531–548 (2004)
Kingman, J.F.C.: Random secants of a convex body. J. Appl. Probab. 6, 660–672 (1969)
Reitzner, M.: Random polytopes and the Efron–Stein jackknife inequality. Ann. Probab. 31, 2136–2166 (2003)
Reitzner, M.: Random polytopes. In: Kendall, W., Molchanov, I. (eds.) New Perspectives in Stochastic Geometry, pp. 45–76. Oxford University Press, Oxford (2010)
Rényi, A., Sulanke, R.: Über die konvexe Hülle von \(n\) zufällig gewählten Punkten. Z. Wahrsch. Verw. Geb. 2, 75–84 (1963)
Rényi, A., Sulanke, R.: Über die konvexe Hülle von \(n\) zufällig gewählten Punkten II. Z. Wahrsch. Verw. Geb. 3, 138–147 (1964)
Schneider, R., Weil, W.: Stochastic and Integral Geometry. Probability and Its Applications (New York). Springer, Berlin (2008)
Zinani, A.: The expected volume of a tetrahedron whose vertices are chosen at random in the interior of a cube. Monatsh. Math. 139, 341–348 (2003)
Acknowledgments
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.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Beermann, M., Reitzner, M. Beyond the Efron–Buchta Identities: Distributional Results for Poisson Polytopes. Discrete Comput Geom 53, 226–244 (2015). https://doi.org/10.1007/s00454-014-9649-7
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00454-014-9649-7