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Approximation Properties of Random Polytopes Associated with Poisson Hyperplane Processes

Part of: Geometric probability and stochastic geometry

Published online by Cambridge University Press:  22 February 2016

Daniel Hug  and
Rolf Schneider
Daniel Hug*
Affiliation:
Karlsruhe Institute of Technology
Rolf Schneider*
Affiliation:
Albert-Ludwigs-Universität Freiburg
*
∗ Postal address: Department of Mathematics, Karlsruhe Institute of Technology, D-76128 Karlsruhe, Germany. Email address: daniel.hug@kit.edu
∗∗ Postal address: Mathematisches Institut, Albert-Ludwigs-Universität Freiburg, D-79104 Freiburg, Germany. Email address: rolf.schneider@math.uni-freiburg.de
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Abstract

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We consider a stationary Poisson hyperplane process with given directional distribution and intensity in d-dimensional Euclidean space. Generalizing the zero cell of such a process, we fix a convex body K and consider the intersection of all closed halfspaces bounded by hyperplanes of the process and containing K. We study how well these random polytopes approximate K (measured by the Hausdorff distance) if the intensity increases, and how this approximation depends on the directional distribution in relation to properties of K.

Keywords

Poisson hyperplane processzero polytopeapproximation of convex bodiesdirectional distribution

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 46 , Issue 4 , December 2014 , pp. 919 - 936
Copyright
© Applied Probability Trust 

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