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The Mean Width of Circumscribed Random Polytopes

Published online by Cambridge University Press:  20 November 2018

Károly J. Böröczky  and
Rolf Schneider
Károly J. Böröczky
Affiliation:
Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, Budapest, Hungary, andDepartment of Geometry, Roland Eötvös University, Pázmány Péter sétány 1/C, Budapest, Hungary e-mail: carlos@renyi.hu
Rolf Schneider
Affiliation:
Mathematisches Institut, Albert-Ludwigs-Universität, Freiburg i. Br., Germany e-mail: rolf.schneider@math.uni-freiburg.de
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Abstract

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For a given convex body $K$ in ${{\mathbb{R}}^{d}}$, a random polytope ${{K}^{(n)}}$ is defined (essentially) as the intersection of $n$ independent closed halfspaces containing $K$ and having an isotropic and (in a specified sense) uniform distribution. We prove upper and lower bounds of optimal orders for the difference of the mean widths of ${{K}^{(n)}}$ and $K$ as $n$ tends to infinity. For a simplicial polytope $P$, a precise asymptotic formula for the difference of the mean widths of ${{P}^{(n)}}$ and $P$ is obtained.

Keywords

52A2260D0552A27random polytopemean widthapproximation
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 53 , Issue 4 , 01 December 2010 , pp. 614 - 628
Copyright
Copyright © Canadian Mathematical Society 2010

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