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On the approximation of a ball by random polytopes

Part of: General convexity Geometric probability and stochastic geometry

Published online by Cambridge University Press:  01 July 2016

K.-H. Küfer
K.-H. Küfer*
Affiliation:
University of Kaiserslautern
*
* Postal address: Department of Mathematics, University of Kaiserslautern, Erwin-Schrödinger-Straße, Post Box 3049, D-67663 Kaiserslautern, Germany.
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Abstract

Let be a sequence of independent and identically distributed random vectors drawn from the d-dimensional unit ball Bd and let Xn be the random polytope generated as the convex hull of a1,· ··, an. Furthermore, let Δ(Xn): = Vol (BdXn) be the volume of the part of the ball lying outside the random polytope. For uniformly distributed ai and 2 we prove that the limiting distribution of Δ(Xn)/Ε (Δ (Xn)) for n → ∞ (satisfies a 0–1 law. In particular, we show that Var for n → ∞. We provide analogous results for spherically symmetric distributions in Bd with regularly varying tail. In addition, we indicate similar results for the surface area and the number of facets of Xn.

Keywords

STOCHASTIC APPROXIMATIONCONVEX HULLVARIANCELIMITING DISTRIBUTION

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Secondary: 52A22: Random convex sets and integral geometry
Type
Research Article
Information
Advances in Applied Probability , Volume 26 , Issue 4 , December 1994 , pp. 876 - 892
Copyright
Copyright © Applied Probability Trust 1994 

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