Large faces in Poisson hyperplane mosaics

Abstract

A generalized version of a well-known problem of D. G. Kendall states that the zero cell of a stationary Poisson hyperplane tessellation in ℝ^d, under the condition that it has large volume, approximates with high probability a certain definite shape, which is determined by the directional distribution of the underlying hyperplane process. This result is extended here to typical k-faces of the tessellation, for k∈{2, …, d−1}. This requires the additional condition that the direction of the face be in a sufficiently small neighbourhood of a given direction.