Stationary Voronoi tessellations

Abstract

A random tessellation is said to be stationary if its distribution is invariant under translations in ℝ. Assuming stationarity it is possible to define what is meant by a typical cell and a typical k-facet of the tessellation. The objective in this chapter is to formalize these concepts and study their relationships for stationary Voronoi and Delaunay tessellations. In fact all results presented for stationary Voronoi tessellations hold as well for arbitrary normal stationary tessellations with convex cells. ‘Normality’ in this context means that every k-facet lies in the boundaries of exactly d−k+1 cells, k = 0,…, d−1. Many real-life non-artificial tessellations for d = 1, 2, 3 possess this property. Indeed Voronoi tessellations in general quadratic position are normal, cf. Proposition 2.1.1. Though Delaunay tessellations are not, many results for Voronoi tessellations carry over because of the duality.