On the measure of Voronoi cells

TL;DR

This paper investigates the distribution of the measure of typical Voronoi cells generated by random points in high-dimensional space, revealing that the distribution becomes more concentrated as the dimension increases.

Contribution

It establishes the asymptotic distribution of Voronoi cell measures, independent of the point location and density, and analyzes how this distribution concentrates in high dimensions.

Findings

Distribution of cell measure is independent of location and density

Variance of the measure converges to zero exponentially fast in dimension

Distribution becomes more concentrated as dimension increases

Abstract

$n$ independent random points drawn from a density $f$ in $R^d$ define a random Voronoi partition. We study the measure of a typical cell of the partition. We prove that the asymptotic distribution of the probability measure of the cell centered at a point $x \in R^d$ is independent of $x$ and the density $f$. We determine all moments of the asymptotic distribution and show that the distribution becomes more concentrated as $d$ becomes large. In particular, we show that the variance converges to zero exponentially fast in $d$. %We also study the measure of the largest cell of the partition. %{\red We also obtain a density-free bound for the rate of convergence of the diameter of a typical Voronoi cell.