The polytopes in a Poisson hyperplane tessellation

TL;DR

This paper proves that in a stationary Poisson hyperplane tessellation, every combinatorial type of simple d-polytope appears with positive density, strengthening previous results about their infinite occurrence.

Contribution

It establishes that all combinatorial types of simple polytopes occur with positive density in the tessellation, not just infinitely often.

Findings

Every combinatorial type appears with positive density.

The result holds under mild conditions on the directional distribution.

It extends previous infinite occurrence results to positive density.

Abstract

For a stationary Poisson hyperplane tessellation $X$ in ${\mathbb R}^d$, whose directional distribution satisfies some mild conditions (which hold in the isotropic case, for example), it was recently shown that with probability one every combinatorial type of a simple $d$-polytope is realized infinitely often by the polytopes of $X$. This result is strengthened here: with probability one, every such combinatorial type appears among the polytopes of $X$ not only infinitely often, but with positive density.