Poisson--Voronoi approximation

TL;DR

This paper analyzes the approximation of convex sets using Poisson--Voronoi tessellations, providing expectation, variance, and concentration bounds for the volume difference and symmetric difference.

Contribution

It introduces new variance estimates and concentration inequalities for Poisson--Voronoi approximations of convex sets, advancing understanding of their probabilistic properties.

Findings

Explicit formulas for the expectation of volume difference and symmetric difference.

New variance bounds derived from a jackknife inequality for Poisson functionals.

Concentration inequalities established using Azuma's inequality.

Abstract

Let $X$ be a Poisson point process and $K\subset\mathbb{R}^d$ a measurable set. Construct the Voronoi cells of all points $x\in X$ with respect to $X$, and denote by $v_X(K)$ the union of all Voronoi cells with nucleus in $K$. For $K$ a compact convex set the expectation of the volume difference $V(v_X(K))-V(K)$ and the symmetric difference $V(v_X(K)\triangle K)$ is computed. Precise estimates for the variance of both quantities are obtained which follow from a new jackknife inequality for the variance of functionals of a Poisson point process. Concentration inequalities for both quantities are proved using Azuma's inequality.