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


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.


Central limit theorem Random polytope Surface body Stochastic geometry 

Mathematics Subject Classification

52A22 60D05 60F05 



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.


  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

Personalised recommendations