Cells with a Given Number of Facets

The complementary theorem of Miles says, very roughly, that, under an equivariance assumption, the suitably measured ‘size’ of certain random sets constructed from a Poisson process, under a suitable cardinality condition, is gamma distributed and is independent of the ‘shape’. Miles has formulated his theorem in different versions, and there are later generalization by various authors. In this chapter, we consider the complementary theorem only for stationary Poisson hyperplane tessellations. It yields, especially, that the suitably measured size, namely the hitting functional, of the typical cell, under the condition that the typical cell has a given number of facets, has a gamma distribution, and that this size is independent of the ‘shape’ of the typical cell. Similar results hold for the zero cell. It is essential here that ‘size’ is measured in a very special way, namely by the hitting functional. In the last section, we reproduce estimates for the probability that the typical cell of a stationary Poisson hyperplane tessellation has a large number of facets.