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Approximation Properties of Random Polytopes Associated with Poisson Hyperplane Processes

Published online by Cambridge University Press:  22 February 2016

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: [email protected]
∗∗ Postal address: Mathematisches Institut, Albert-Ludwigs-Universität Freiburg, D-79104 Freiburg, Germany. Email address: [email protected]
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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.

Type
Stochastic Geometry and Statistical Applications
Copyright
© Applied Probability Trust 

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