Title: Small cells in a Poisson hyperplane tessellation,
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.