Poisson-Voronoi tessellations in three-dimensional hyperbolic spaces

Abstract

We study Poisson-Voronoi tessellations in three-dimensional hyperbolic spaces, and give explicit expressions for mean surface area, mean perimeter length, and mean number of vertices of their cells. Furthermore we compare these mean characteristics with those for Poisson-Voronoi tessellations in three-dimensional Euclidean spaces. It is shown that, as the absolute value of the curvature of hyperbolic spaces increases from zero to infinity, these mean characteristics increase monotonically from those for the Euclidean case to infinity.