# Small cells in a Poisson hyperplane tessellation

## TL;DR

This paper systematically studies small cells in Poisson hyperplane tessellations across arbitrary dimensions, revealing their tendency to minimize facets and their shape distribution dependence on size and directionality.

## Contribution

It provides the first comprehensive analysis of small cells in high-dimensional Poisson hyperplane tessellations with arbitrary size functionals and directional distributions.

## Key findings

- Small cells tend to minimize the number of facets.
- Small cells have a non-degenerate limit shape distribution.
- Certain directional distributions lead to cells with small inradius not minimizing facets.

## Abstract

Until now, little was known about properties of small cells in a Poisson hyperplane tessellation. The few existing results were either heuristic or applying only to the two dimensional case and for very specific size functionals and directional distributions. This paper fills this gap by providing a systematic study of small cells in a Poisson hyperplane tessellation of arbitrary dimension, arbitrary directional distribution $\varphi$ and with respect to an arbitrary size functional $\Sigma$. More precisely, we investigate the distribution of the typical cell $Z$, conditioned on the event $\{\Sigma(Z)<a\}$, where $a\to0$ and $\Sigma$ is a size functional, i.e. a functional on the set of convex bodies which is continuous, not identically zero, homogeneous of degree $k>0$, and increasing with respect to set inclusion. We focus on the number of facets and the shape of such small cells. We show in various general settings that small cells tend to minimize the number of facets and that they have a non degenerated limit shape distribution which depends on the size $\Sigma$ and the directional distribution. We also exhibit a class of directional distribution for which cells with small inradius do not tend to minimize the number of facets.