On the cells in a stationary Poisson hyperplane mosaic

TL;DR

This paper investigates the properties of cells in a stationary Poisson hyperplane mosaic, showing that the cells' shapes are dense in convex bodies, all simple polytope types occur infinitely often, and analyzing the typical cell distribution.

Contribution

It establishes that the set of cells in such mosaics is dense in convex bodies and all simple polytope types appear infinitely often, under mild directional distribution conditions.

Findings

Cells are dense in the space of convex bodies.

All combinatorial types of simple d-polytopes occur infinitely often.

Distribution of the typical cell is characterized.

Abstract

Let $X$ be the mosaic generated by a stationary Poisson hyperplane process $\hat X$ in ${\mathbb R}^d$. Under some mild conditions on the spherical directional distribution of $\hat X$ (which are satisfied, for example, if the process is isotropic), we show that with probability one the set of cells ($d$-polytopes) of $X$ has the following properties. The translates of the cells are dense in the space of convex bodies. Every combinatorial type of simple $d$-polytopes is realized infinitely often by the cells of $X$. A further result concerns the distribution of the typical cell.