Cells with many facets in a Poisson hyperplane tessellation

TL;DR

This paper analyzes the distribution of the number of facets of a typical cell in a Poisson hyperplane tessellation, revealing bounds and asymptotic behaviors under certain conditions.

Contribution

It provides new bounds and asymptotic results for the distribution of facets in Poisson hyperplane tessellations, utilizing geometric estimates and the Complementary Theorem.

Findings

Bounded the quantity involving the probability of large number of facets.

Established that large facets imply a bounded away from zero isoperimetric ratio.

Derived tail estimates for the $\

Abstract

Let $Z$ be the typical cell of a stationary Poisson hyperplane tessellation in $\mathbb{R}^d$. The distribution of the number of facets $f(Z)$ of the typical cell is investigated. It is shown, that under a well-spread condition on the directional distribution, the quantity $n^{\frac{2}{d-1}}\sqrt[n]{\mathbb{P}(f(Z)=n)}$ is bounded from above and from below. When $f(Z)$ is large, the isoperimetric ratio of $Z$ is bounded away from zero with high probability. These results rely on one hand on the Complementary Theorem which provides a precise decomposition of the distribution of $Z$ and on the other hand on several geometric estimates related to the approximation of polytopes by polytopes with fewer facets. From the asymptotics of the distribution of $f(Z)$, tail estimates for the so-called $\Phi$ content of $Z$ are derived as well as results on the conditional distribution of $Z$ when its $\Phi$ content is large.