Advances in Applied Probability

Faces with given directions in anisotropic Poisson hyperplane mosaics

Daniel Hug and Rolf Schneider

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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.

Article information

Adv. in Appl. Probab., Volume 43, Number 2 (2011), 308-321.

First available in Project Euclid: 21 June 2011

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]

Poisson hyperplane tessellation volume-weighted typical face typical face conditional distribution Blaschke body shape


Hug, Daniel; Schneider, Rolf. Faces with given directions in anisotropic Poisson hyperplane mosaics. Adv. in Appl. Probab. 43 (2011), no. 2, 308--321.

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