Cells with many facets in a Poisson hyperplane tessellation

Let Z be the typical cell of a stationary Poisson hyperplane tessellation in Rd. 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 n2d−1P(f(Z)=n)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.