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Stochastical approximation of smooth convex bodies

Part of:General convexity

Published online by Cambridge University Press:  26 February 2010

Matthias Reitzner
Matthias Reitzner
Affiliation:
Institut für Diskrete Mathematik und Geometrie, Technische Universität Wien, Wiedner Hauptstrasse 8/104, A-1040 Wien, Austria
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Abstract

A random polytope is the convex hull of n random points in the interior of a convex body K. The expectation of the ith intrinsic volume of a random polytope as n → ∞ is investigated. It is proved that, for convex bodies of differentiability class Kk+1, precise asymptotic expansions for these expectations exist. The proof makes essential use of a refinement of Crofton's boundary theorem.

MSC classification

Secondary:52A22: Random convex sets and integral geometry

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Type
Research Article
Information
Mathematika , Volume 51 , Issue 1-2 , December 2004, pp. 11 - 29
Copyright
Copyright © University College London 2004

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