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Discrete & Computational Geometry

, Volume 59, Issue 4, pp 990–1000 | Cite as

Central Limit Theorem for the Volume of Random Polytopes with Vertices on the Boundary

  • Christoph ThäleEmail author
Article

Abstract

Given a convex body K with smooth boundary \(\partial K\), select a fixed number n of uniformly distributed random points from \(\partial K\). The convex hull \(K_n\) of these points is a random polytope having all its vertices on the boundary of K. The closeness of the volume of \(K_n\) to a Gaussian random variable is investigated in terms of the Kolmogorov distance by combining a version of Stein’s method with geometric estimates for the surface body of K.

Keywords

Central limit theorem Random polytope Surface body Stochastic geometry 

Mathematics Subject Classification

52A22 60D05 60F05 

Notes

Acknowledgements

I am grateful to Julian Grote (Bochum) for stimulating discussions and helpful comments. I also thank two anonymous referees for careful reading and their suggestions.

References

  1. 1.
    Bőrőczky, K.J., Fodor, F., Hug, D.: Intrinsic volumes of random polytopes with vertices on the boundary of a convex body. Trans. Am. Math. Soc. 365(2), 785–809 (2013)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Buchta, C., Müller, J., Tichy, R.F.: Stochastical approximation of convex bodies. Math. Ann. 271(2), 225–235 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Lachièze-Rey, R., Peccati, G.: New Kolmogorov Berry–Esseen bounds for functionals of binomial point processes. Ann. Appl. Probab. (accepted)Google Scholar
  4. 4.
    Last, G., Peccati, G., Schulte, M.: Normal approximation on Poisson spaces: Mehler’s formula, second order Poincaré inequalities and stabilization. Probab. Theory Relat. Fields 165(3–4), 667–723 (2016)CrossRefzbMATHGoogle Scholar
  5. 5.
    Reitzner, M.: Random points on the boundary of smooth convex bodies. Trans. Am. Math. Soc. 354(6), 2243–2278 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Reitzner, M.: Random polytopes and the Efron–Stein jackknife inequality. Ann. Probab. 31(4), 2136–2166 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Reitzner, M.: Central limit theorems for random polytopes. Probab. Theory Relat. Fields 133(4), 483–507 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Reitzner, M.: Random polytopes. In: Kendall, W.S., Molchanov, I. (eds.) New Perspectives in Stochastic Geometry, pp. 45–76. Oxford University Press, Oxford (2010)Google Scholar
  9. 9.
    Richardson, R.M., Vu, V.H., Wu, L.: Random inscribing polytopes. Eur. J. Comb. 28(8), 2057–2071 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Richardson, R.M., Vu, V.H., Wu, L.: An inscribing model for random polytopes. Discrete Comput. Geom. 39(1–3), 469–499 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Schütt, C., Werner, E.: Polytopes with vertices chosen randomly from the boundary of a convex body. In: Milman, V.D., Schechtman, G. (eds.) Geometric Aspects of Functional Analysis. Lecture Notes in Mathematics, vol. 1807, pp. 241–422. Springer, Berlin (2003)CrossRefGoogle Scholar
  12. 12.
    Schütt, C., Werner, E.: Surface bodies and \(p\)-affine surface area. Adv. Math. 187(1), 98–145 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Vu, V.H.: Sharp concentration of random polytopes. Geom. Funct. Anal. 15(6), 1284–1318 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Vu, V.: Central limit theorems for random polytopes in a smooth convex set. Adv. Math. 207(1), 221–243 (2006)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Faculty of MathematicsRuhr University BochumBochumGermany

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