On variances of partial volumes of the typical cell of a Poisson-Voronoi tessellation and large-dimensional volume degeneracy

Abstract

For a typical cell of a homogeneous Poisson-Voronoi tessellation in ℝ d , it is shown that the variance of the volume of the intersection of the typical cell with any measurable subset of ℝ d is bounded by the variance of the volume of the typical cell. It is also shown that the variance of the volume of the intersection of the typical cell with a translation of itself is bounded by four times the variance of the volume of the typical cell. These bounds are applied to show large-dimensional volume degeneracy as d tends to ∞. An extension to the kth nearest-point Poisson-Voronoi tessellation for k ≥ 2 is also considered.