Faces with given directions in anisotropic Poisson hyperplane mosaics

Abstract

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.