Advances in Applied Probability

Large deviation probabilities for the number of vertices of random polytopes in the ball

Pierre Calka and Tomasz Schreiber

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In this paper we establish large deviation results on the number of extreme points of a homogeneous Poisson point process in the unit ball of Rd. In particular, we deduce an almost-sure law of large numbers in any dimension. As an auxiliary result we prove strong localization of the extreme points in an annulus near the boundary of the ball.

Article information

Adv. in Appl. Probab. Volume 38, Number 1 (2006), 47-58.

First available in Project Euclid: 1 April 2006

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]
Secondary: 60F10: Large deviations

Convex hull covering of the sphere large deviation measure concentration random polytope


Calka, Pierre; Schreiber, Tomasz. Large deviation probabilities for the number of vertices of random polytopes in the ball. Adv. in Appl. Probab. 38 (2006), no. 1, 47--58. doi:10.1239/aap/1143936139.

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  • Bárány, I. (1992). Random polytopes in smooth convex bodies. Mathematika 39, 81--92.
  • Bárány, I. and Larman, D. G. (1988). Convex bodies, economic cap coverings, random polytopes. Mathematika 35, 274--291.
  • Buchta, C. and Müller, J. (1984). Random polytopes in a ball. J. Appl. Prob. 21, 753--762.
  • Calka, P. (2002). The distributions of the smallest disks containing the Poisson--Voronoi typical cell and the Crofton cell in the plane. Adv. Appl. Prob. 34, 702--717.
  • Calka, P. and Schreiber, T. (2005). Limit theorems for the typical Poisson--Voronoi cell and the Crofton cell with a large inradius. Ann. Prob. 33, 1625--1642.
  • Efron, B. (1965). The convex hull of a random set of points. Biometrika 52, 331--343.
  • Groeneboom, P. (1988). Limit theorems for convex hulls. Prob. Theory Relat. Fields 79, 327--368.
  • Ledoux, M. (2001). The Concentration of Measure Phenomenon (Math. Surveys Monogr. 89). American Mathematical Society, Providence, RI.
  • Massé, B. (2000). On the LLN for the number of vertices of a random convex hull. Adv. Appl. Prob. 32, 675--681.
  • Reitzner, M. (2003). Random polytopes and the Efron--Stein jackknife inequality. Ann. Prob. 31, 2136--2166.
  • Rényi, A. and Sulanke, R. (1963). Über die konvexe Hülle von $n$ zufällig gewählten Punkten. Z. Wahrschein- lichkeitsth. 2, 75--84.
  • Schreiber, T. (2003). A note on large deviation probabilities for volumes of unions of random closed sets. Submitted. Available at
  • Schütt, C. (1994). Random polytopes and affine surface area. Math. Nachr. 170, 227--249.
  • Wieacker, J. A. (1978). Einige Probleme der polyedrischen Approximation. Doctoral Thesis, Universität Freiburg.