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Intrinsic volumes of inscribed random polytopes in smooth convex bodies

Published online by Cambridge University Press:  01 July 2016

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: [email protected]
∗∗ Postal address: Department of Geometry, University of Szeged, Aradi vértanúk tere 1, H-6720 Szeged, Hungary. Email address: [email protected]
∗∗∗ Postal address: Bolyai Institute, University of Szeged, Aradi vértanúk tere 1, H-6720 Szeged, Hungary. Email address: [email protected]
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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.

Type
Stochastic Geometry and Statistical Applications
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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