Sectional Voronoi tessellations: Characterization and high-dimensional limits

TL;DR

This paper studies the geometric properties of sectional Voronoi tessellations, showing how intersections with affine subspaces relate to different models and demonstrating high-dimensional convergence to Gaussian-Voronoi tessellations.

Contribution

It characterizes the intersections of various Voronoi tessellations with affine subspaces and proves their high-dimensional limits to Gaussian-Voronoi tessellations.

Findings

Intersection of Poisson-Voronoi with affine subspace matches beta-Voronoi distribution.

Detailed analysis of typical cell and face properties.

High-dimensional limit of Poisson-Voronoi intersection converges to Gaussian-Voronoi.

Abstract

The intersections of beta-Voronoi, beta-prime-Voronoi and Gaussian-Voronoi tessellations in $\mathbb{R}^d$ with $\ell$-dimensional affine subspaces, $1\leq \ell\leq d-1$, are shown to be random tessellations of the same type but with different model parameters. In particular, the intersection of a classical Poisson-Voronoi tessellation with an affine subspace is shown to have the same distribution as a certain beta-Voronoi tessellation. The geometric properties of the typical cell and, more generally, typical $k$-faces, of the sectional Poisson-Voronoi tessellation are studied in detail. It is proved that in high dimensions, that is as $d\to\infty$, the intersection of the $d$-dimensional Poison-Voronoi tessellation with an affine subspace of fixed dimension $\ell$ converges to the $\ell$-dimensional Gaussian-Voronoi tessellation.