The Poisson Voronoi Tessellation I. A Basic Identity

This paper gives basic relations between the stationary Poisson point process and the point process of vertices of the corresponding Voronoi tessellation in IRd and of planar sections through it. The results are based on a study of the Palm distribution of the point process of vertices. An identity is given connecting the distribution of a Poisson point process and the Palm distribution with respect to the vertices of the corresponding Voronoi tessellation. Distributional properties for the edges are discussed. Finally, identities are given for characteristics of the “typical” edge and an edge chosen at random emanating from the “typical” vertex.