Advertisement

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
Article

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.

Keywords

Poisson polytope Random polytope Generating function Efron’s identity 

Mathematics Subject Classification

Primary 60D05 Secondary 60G55 52A22 

Notes

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.

References

  1. 1.
    Affentranger, F.: The expected volume of a random polytope in a ball. J. Microsc. 151, 277–287 (1988)CrossRefGoogle Scholar
  2. 2.
    Bárány, I.: Random polytopes in smooth convex bodies. Mathematika 39, 81–92 (1992)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Bárány, I., Reitzner, M.: Poisson polytopes. Ann. Probab. 38, 1507–1531 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Bárány, I., Reitzner, M.: On the variance of random polytopes. Adv. Math. 225, 1986–2001 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Buchta, C.: Stochastische Approximation konvexer Polygone. Z. Wahrsch. Verw. Geb. 67, 283–304 (1984)Google Scholar
  6. 6.
    Buchta, C.: Zufallspolygone in konvexen Vielecken. J. Reine Angew. Math. 347, 212–220 (1984)zbMATHMathSciNetGoogle Scholar
  7. 7.
    Buchta, C.: An identity relating moments of functionals of convex hulls. Discrete Comput. Geom. 33, 125–142 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Buchta, C., Müller, J.: Random polytopes in a ball. J. Appl. Probab. 21, 753–762 (1984)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    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)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    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)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Calka, P., Yukich, J.E.: Variance asymptotics for random polytopes in smooth convex bodies. Probab. Theory Relat. Fields 152, 435–463 (2014)CrossRefMathSciNetGoogle Scholar
  12. 12.
    Doetsch, G.: Handbuch der Laplace-Transformation II. Birkhäuser, Basel, Stuttgart (1955)Google Scholar
  13. 13.
    Efron, B.: The convex hull of a random set of points. Biometrika 52, 331–343 (1965)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    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)Google Scholar
  15. 15.
    Hug, D., Munsonius, G.O., Reitzner, M.: Asymptotic mean values of Gaussian polytopes. Beitr. Algebra Geom. 45, 531–548 (2004)zbMATHMathSciNetGoogle Scholar
  16. 16.
    Kingman, J.F.C.: Random secants of a convex body. J. Appl. Probab. 6, 660–672 (1969)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Reitzner, M.: Random polytopes and the Efron–Stein jackknife inequality. Ann. Probab. 31, 2136–2166 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Reitzner, M.: Random polytopes. In: Kendall, W., Molchanov, I. (eds.) New Perspectives in Stochastic Geometry, pp. 45–76. Oxford University Press, Oxford (2010)Google Scholar
  19. 19.
    Rényi, A., Sulanke, R.: Über die konvexe Hülle von \(n\) zufällig gewählten Punkten. Z. Wahrsch. Verw. Geb. 2, 75–84 (1963)CrossRefzbMATHGoogle Scholar
  20. 20.
    Rényi, A., Sulanke, R.: Über die konvexe Hülle von \(n\) zufällig gewählten Punkten II. Z. Wahrsch. Verw. Geb. 3, 138–147 (1964)CrossRefzbMATHGoogle Scholar
  21. 21.
    Schneider, R., Weil, W.: Stochastic and Integral Geometry. Probability and Its Applications (New York). Springer, Berlin (2008)Google Scholar
  22. 22.
    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)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of OsnabrueckOsnabrueckGermany

Personalised recommendations