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Faces with given directions in anisotropic Poisson hyperplane mosaics

Part of: Geometric probability and stochastic geometry

Published online by Cambridge University Press:  01 July 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, Eckerstr. 1, D-79104 Freiburg, Germany. Email address: rolf.schneider@math.uni-freiburg.de
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Abstract

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For stationary Poisson hyperplane tessellations in d-dimensional Euclidean space and a dimension k ∈ {1, …, d}, we investigate the typical k-face and the weighted typical k-face (weighted by k-dimensional volume), without isotropy assumptions on the tessellation. The case k = d concerns the previously studied typical cell and zero cell, respectively. For k < d, we first find the conditional distribution of the typical k-face or weighted typical k-face, given its direction. Then we investigate how the shapes of the faces are influenced by assumptions of different types: either via containment of convex bodies of given volume (including a new result for k = d), or, for weighted typical k-faces, in the spirit of D. G. Kendall's asymptotic problem, suitably generalized. In all these results on typical or weighted typical k-faces with given direction space L, the Blaschke body of the section process of the underlying hyperplane process with L plays a crucial role.

Keywords

Poisson hyperplane tessellationvolume-weighted typical facetypical faceconditional distributionBlaschke bodyshape

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 43 , Issue 2 , June 2011 , pp. 308 - 321
Copyright
Copyright © Applied Probability Trust 2011 

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