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

  • Pierre Calka (a1) and Tomasz Schreiber (a2)

Abstract

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 R d . 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.

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Copyright

Corresponding author

Postal address: Université René Descartes Paris 5, MAP5, UFR Math-Info, 45 rue des Saints-Pères, 75270 Paris Cedex 06, France. Email address: pierre.calka@math-info.univ-paris5.fr
∗∗ Postal address: Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, ul. Chopina 12/18, 87-100 Toruń, Poland. Email address: tomeks@mat.uni.torun.pl

References

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[4] 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, 702717.
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[12] Schreiber, T. (2003). A note on large deviation probabilities for volumes of unions of random closed sets. Submitted. Available at http://www.mat.uni.torun.pl/preprints.
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[14] Wieacker, J. A. (1978). Einige Probleme der polyedrischen Approximation. , Universität Freiburg.

Keywords

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

  • Pierre Calka (a1) and Tomasz Schreiber (a2)

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