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 ℝd, it is shown that the variance of the volume of the intersection of the typical cell with any measurable subset of ℝd 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.

On the Volume of the Zero Cell of a Class of Isotropic Poisson Hyperplane Tessellations

“…In particular, there exist only a few results on random tessellations in high dimensions, that is, with focus on asymptotic aspects as the dimension goes to ∞. Recently, the typical cell of a stationary Poisson-Voronoi tessellation in high dimensions has been studied in [1], [14], and [27]. Alishahi and Sharifitabar [1] investigated the asymptotic behaviour of the volume and the shape of the typical cell of a stationary Poisson-Voronoi tessellation as the dimension n of the space goes to ∞.…”

On the Volume of the Zero Cell of a Class of Isotropic Poisson Hyperplane Tessellations

“…In particular, there exist only a few results on random tessellations in high dimensions, that is, with focus on asymptotic aspects as the dimension goes to infinity. Recently, the typical cell of a stationary Poisson-Voronoi tessellation in high dimensions has been studied in [1], [14] and [27]. Alishahi and Sharifitabar [1] investigate the asymptotic behaviour of the volume and the shape of the typical cell of a stationary Poisson-Voronoi tessellation as the dimension n of the space goes to infinity.…”