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Intrinsic volumes of inscribed random polytopes in smooth convex bodies

Part of: General convexity Geometric probability and stochastic geometry

Published online by Cambridge University Press:  01 July 2016

I. Bárány ,
F. Fodor  and
V. Vígh
I. Bárány*
Affiliation:
Alfréd Rényi Institute of Mathematics and University College London
F. Fodor*
Affiliation:
University of Szeged and University of Calgary
V. Vígh*
Affiliation:
University of Szeged
*
∗ Postal address: Department of Mathematics, University College London, Gower Street, London, WC1E 6BT, UK. Email address: barany@renyi.hu
∗∗ Postal address: Department of Geometry, University of Szeged, Aradi vértanúk tere 1, H-6720 Szeged, Hungary. Email address: fodorf@math.u-szeged.hu
∗∗∗ Postal address: Bolyai Institute, University of Szeged, Aradi vértanúk tere 1, H-6720 Szeged, Hungary. Email address: vigvik@math.u-szeged.hu
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Abstract

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Let K be a d-dimensional convex body with a twice continuously differentiable boundary and everywhere positive Gauss-Kronecker curvature. Denote by Kn the convex hull of n points chosen randomly and independently from K according to the uniform distribution. Matching lower and upper bounds are obtained for the orders of magnitude of the variances of the sth intrinsic volumes Vs(Kn) of Kn for s ∈ {1,…,d}. Furthermore, strong laws of large numbers are proved for the intrinsic volumes of Kn. The essential tools are the economic cap covering theorem of Bárány and Larman, and the Efron-Stein jackknife inequality.

Keywords

Convex bodyintrinsic volumeuniform random polytopevariancestrong law of large numbers

MSC classification

Primary: 52A22: Random convex sets and integral geometry 60D05: Geometric probability and stochastic geometry
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 42 , Issue 3 , September 2010 , pp. 605 - 619
Copyright
Copyright © Applied Probability Trust 2010 

Footnotes

Supported by Hungarian OTKA grant 60427.

Supported by Hungarian OTKA grants 68398 and 75016, and by the János Bolyai Research Scholarship of the Hungarian Academy of Sciences.

Supported by Hungarian OTKA grant 75016.

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