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THE VOLUME OF RANDOM POLYTOPES CIRCUMSCRIBED AROUND A CONVEX BODY

Part of: General convexity Geometric probability and stochastic geometry

Published online by Cambridge University Press:  22 June 2015

Ferenc Fodor ,
Daniel Hug  and
Ines Ziebarth
Ferenc Fodor
Affiliation:
Department of Geometry, Bolyai Institute, University of Szeged, Aradi vértanúk tere 1, 6720 Szeged, Hungary Department of Mathematics and Statistics, University of Calgary, Canada email [email protected]
Daniel Hug
Affiliation:
Karlsruhe Institute of Technology, Department of Mathematics, D-76128 Karlsruhe, Germany email [email protected]
Ines Ziebarth
Affiliation:
Karlsruhe Institute of Technology, Department of Mathematics, D-76128 Karlsruhe, Germany email [email protected]
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Abstract

Let $K$ be a convex body in $\mathbb{R}^{d}$ which slides freely in a ball. Let $K^{(n)}$ denote the intersection of $n$ closed half-spaces containing $K$ whose bounding hyperplanes are independent and identically distributed according to a certain prescribed probability distribution. We prove an asymptotic formula for the expectation of the difference of the volumes of $K^{(n)}$ and $K$, and an asymptotic upper bound on the variance of the volume of $K^{(n)}$. We obtain these asymptotic formulas by proving results for weighted mean width approximations of convex bodies that admit a rolling ball by inscribed random polytopes and then using polar duality to convert them into statements about circumscribed random polytopes.

MSC classification

Primary: 52A22: Random convex sets and integral geometry
Secondary: 60D05: Geometric probability and stochastic geometry 52A27: Approximation by convex sets
Type
Research Article
Information
Mathematika , Volume 62 , Issue 1 , 2016 , pp. 283 - 306
Copyright
Copyright © University College London 2015 

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