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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: [email protected]
∗∗ Postal address: Mathematisches Institut, Albert-Ludwigs-Universität Freiburg, Eckerstr. 1, D-79104 Freiburg, Germany. Email address: [email protected]
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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 

References

Baumstark, V. and Last, G. (2007). Some distributional results for Poisson–Voronoi tessellations. Adv. Appl. Prob. 39, 16–40.CrossRefGoogle Scholar
Hug, D. and Schneider, R. (2007). Asymptotic shapes of large cells in random tessellations. Geom. Funct. Anal. 17, 156–191.Google Scholar
Hug, D. and Schneider, R. (2007). Typical cells in Poisson hyperplane tessellations. Discrete Comput. Geom. 38, 305–319.Google Scholar
Hug, D. and Schneider, R. (2010). Large faces in Poisson hyperplane mosaics. Ann. Prob. 38, 1320–1344.CrossRefGoogle Scholar
Hug, D., Reitzner, M. and Schneider, R. (2004). The limit shape of the zero cell in a stationary Poisson hyperplane tessellation. Ann. Prob. 32, 1140–1167.CrossRefGoogle Scholar
Hug, D., Reitzner, M. and Schneider, R. (2004). Large Poisson–Voronoi cells and Crofton cells. Adv. Appl. Prob. 36, 667–690.Google Scholar
Kovalenko, I. N. (1997). Proof of David Kendall's conjecture concerning the shape of large random polygons. Cybernet. Systems Anal. 33, 461–467.CrossRefGoogle Scholar
Miles, R. E. (1964). Random polygons determined by random lines in a plane. II. Proc. Nat. Acad. Sci. USA 52, 1157–1160.CrossRefGoogle Scholar
Molchanov, I. (2005). Theory of Random Sets. Springer, London.Google Scholar
Møller, J. (1989). Random tessellations in R d . Adv. Appl. Prob. 21, 37–73.Google Scholar
Schneider, R. (1993). Convex Bodies: the Brunn–Minkowski Theory. Cambridge University Press.Google Scholar
Schneider, R. (2009). Weighted faces of Poisson hyperplane tessellations. Adv. Appl. Prob. 41, 682–694.CrossRefGoogle Scholar
Schneider, R. and Weil, W. (2008). Stochastic and Integral Geometry. Springer, Berlin.Google Scholar
Stoyan, D., Kendall, W. S. and Mecke, J. (1995). Stochastic Geometry and Its Applications, 2nd edn. John Wiley, Chichester.Google Scholar