Cells with many facets in a Poisson hyperplane tessellation

Gilles Bonnet (1) , Pierre Calka (2) , Matthias Reitzner (1)

1 Institut für Mathematik
2 LMRS - Laboratoire de Mathématiques Raphaël Salem

Let Z be the typical cell of a stationary Poisson hyperplane tessellation in 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 2 d−1 n 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 Φ content of Z are derived as well as results on the conditional distribution of Z when its Φ content is large.

Keywords: Poisson hyperplane tessellation; random polytopes; typical cell; directional distribution; Complementary Theorem; DG Kendall's problem; shape distribution

Domains: Probability [math.PR]

Gilles Bonnet, Pierre Calka, Matthias Reitzner. Cells with many facets in a Poisson hyperplane tessellation. Advances in Mathematics, 2018, 324, pp.203-240. ⟨hal-01364783⟩