Large faces in Poisson hyperplane mosaics

TL;DR

This paper extends Kendall's problem by showing that large faces in a Poisson hyperplane mosaic tend to approximate a specific shape, depending on the hyperplane directions, especially when the face directions are near a fixed orientation.

Contribution

It generalizes Kendall's shape approximation result from the zero cell to typical k-faces, under directional proximity conditions.

Findings

Large faces approximate a definite shape with high probability.

Shape approximation depends on the directional distribution of hyperplanes.

Results extend Kendall's problem to higher-dimensional faces.

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 ${\mathbb{R}}^d$, 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 $k\in\{2,...,d-1\}$. This requires the additional condition that the direction of the face be in a sufficiently small neighbourhood of a given direction.