Probability Theory and Related Fields

, Volume 133, Issue 4, pp 483–507 | Cite as

Central limit theorems for random polytopes

Article

Abstract

Let K be a smooth convex set. The convex hull of independent random points in K is a random polytope. Central limit theorems for the volume and the number of i dimensional faces of random polytopes are proved as the number of random points tends to infinity. One essential step is to determine the precise asymptotic order of the occurring variances.

Key words or phrases

Random polytopes CLT Approximation of convex bodies Dependency graph 

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References

  1. 1.
    Alagar, V.S.: On the distribution of a random triangle. J. Appl. Probability 14, 284–297 (1977)Google Scholar
  2. 2.
    Avram, F., Bertsimas, D.: On central limit theorems in geometrical probability. Ann. Appl. Probab. 3, 1033–1046 (1993)Google Scholar
  3. 3.
    Baldi, P., Rinott, Y.: On normal approximations of distributions in terms of dependency graphs. Ann. Probab. 17, 1646–1650 (1989)Google Scholar
  4. 4.
    Bárány, I.: Random polytopes in smooth convex bodies. Mathematika 39, 81–92 (1992)Google Scholar
  5. 5.
    Bárány, I.: Sylvester's question: the probability that n points are in convex position. Ann. Probab. 27, 2020–2034 (1999)CrossRefGoogle Scholar
  6. 6.
    Bárány, I.: A note on Sylvester's four-point problem. Studia Sci. Math. Hungar. 38, 73–77 (2001)Google Scholar
  7. 7.
    Bárány, I., Buchta, C.: Random polytopes in a convex polytope, independence of shape, and concentration of vertices. Math. Ann. 297, 467–497 (1993)CrossRefGoogle Scholar
  8. 8.
    Bárány, I., Rote, G., Steiger, W., Zhang, C.-H.: A central limit theorem for convex chains in the square. Discrete Comput. Geom. 23, 35–50 (2000)Google Scholar
  9. 9.
    Buchta, C.: An identity relating moments of functionals of convex hulls. Discrete Comput. Geom. 33, 125–142 (2005)CrossRefGoogle Scholar
  10. 10.
    Cabo, A.J., Groeneboom, P.: Limit theorems for functionals of convex hulls. Probab. Theory Relat. Fields 100, 31–55 (1994)CrossRefGoogle Scholar
  11. 11.
    Efron, B.: The convex hull of a random set of points. Biometrika 52, 331–343 (1965)Google Scholar
  12. 12.
    Finch, S., Hueter, I.: Random convex hulls: a variance revisited. Adv. Appl. Probab. 36, 981–986 (2004)CrossRefGoogle Scholar
  13. 13.
    Groeneboom, P.: Limit theorems for convex hulls. Probab. Theory Relat. Fields 79, 327–368 (1988)CrossRefGoogle Scholar
  14. 14.
    Gruber, P.M.: Comparisons of best and random approximation of convex bodies by polytopes. Rend. Circ. Mat. Palermo, II. Ser., Suppl. 50, 189–216 (1997)Google Scholar
  15. 15.
    Henze, N.: Random triangles in convex regions. J. Appl. Prob. 20, 111–125 (1983)Google Scholar
  16. 16.
    Hsing, T.: On the asymptotic distribution of the area outside a random convex hull in a disk. Ann. Appl. Probab. 4, 478–493 (1994)Google Scholar
  17. 17.
    Hueter, I.: Limit theorems for the convex hull of random points in higher dimensions. Trans. Am. Math. Soc. 351, 4337–4363 (1999)CrossRefGoogle Scholar
  18. 18.
    Hüsler, J.: On the convex hull of dependent random vectors. Rend. Circ. Mat. Palermo, II. Ser. Suppl. 41, 109–117 (1997)Google Scholar
  19. 19.
    McMullen, P.: The maximum numbers of faces of a convex polytope. Mathematika 17, 179–184 (1970)Google Scholar
  20. 20.
    Reitzner, M.: Random points on the boundary of smooth convex bodies. Trans. Am. Math. Soc. 354, 2243–2278 (2002)CrossRefGoogle Scholar
  21. 21.
    Reitzner, M.: Random polytopes and the Efron–Stein jackknife inequality. Ann. Probab. 31, 2136–2166 (2003)CrossRefGoogle Scholar
  22. 22.
    Reitzner, M.: The combinatorial structure of random polytopes. Adv. Math. 191, 178–208 (2005)CrossRefGoogle Scholar
  23. 23.
    Rényi, A., Sulanke, R.: Über die konvexe Hülle von n zufällig gewählten Punkten II. Z. Wahrscheinlichkeitsth. Verw. Geb. 3, 138–147 (1964)CrossRefGoogle Scholar
  24. 24.
    Rinott, Y.: On normal approximation rates for certain sums of dependent random variables. J. Comput. Appl. Math. 55, 135–143 (1994)CrossRefGoogle Scholar
  25. 25.
    Schneider, R.: Random approximation of convex sets. J. Microscopy 151, 211–227 (1988)Google Scholar
  26. 26.
    Schneider, R.: Discrete aspects of stochastic geometry. In: Goodman, J.E., O'Rourke, J. (eds.), Handbook of Discrete and Computational Geometry, Boca Raton: CRC Press 2004 (CRC Press Series on Discrete Mathematics and its Applications), pp. 255–278Google Scholar
  27. 27.
    Schneider, R., Weil, W.: Stochastische Geometrie. Stuttgart: Teubner Skripten zur Mathematischen Stochastik, 2000Google Scholar
  28. 28.
    Stoyan, D., Kendall, W.S., Mecke, J.: Stochastic geometry and its applications. Chichester: Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics, 1987Google Scholar
  29. 29.
    Schütt, C.: Random polytopes and affine surface area. Math. Nachr. 170, 227–249 (1994)Google Scholar
  30. 30.
    Vervaat, W.: Upper bounds for the distance in total variation between the binomial or negative binomial and the Poisson distribution. Statistica Neerlandica 23, 79–86 (1969)Google Scholar
  31. 31.
    Wieacker, J.A.: Einige Probleme der polyedrischen Approximation. Diplomarbeit, Freiburg im Breisgau 1978Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  1. 1.Inst. of Discrete Mathematics and GeometryUniversity of Technology ViennaViennaAustria

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