Cells with many facets in a Poisson hyperplane tessellation

Abstract

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 root P(f(Z)=n is bounded from above and frombelow. 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. (C) 2017 Elsevier Inc. All rights reserved.