Distribution-Independent Density Relations

Much of the following will deal with investigating the average sizes of the polytopes appearing in a stationary random hyperplane tessellation. Here ‘average’ will be made precise by considering, for example, typical cells and k-faces. To measure the sizes, we use a series of functionals, which regard combinatorial or metrical aspects. These functionals are introduced in the following, and the rest of this chapter will then deal with relations between their densities. In the first section, we consider relations for general random face-to-face tessellations, and in the second section we focus on hyperplane tessellations. Apart from some integrability assumptions, the relations do not depend on the distributions of the random tessellations. Correspondingly, the arguments will be more geometric and combinatoric in nature than stochastic.