The Poisson-Voronoi cell around an isolated nucleus

TL;DR

This paper studies the asymptotic shape and properties of a Poisson-Voronoi cell around an isolated nucleus in a planar point process as the intensity increases, revealing convergence to a deterministic convex set and detailed geometric characteristics.

Contribution

It provides a detailed description of the limiting shape of the Voronoi cell and asymptotics of key geometric measures, extending classical results to a new setting with a fixed nucleus.

Findings

Voronoi cell converges to a deterministic convex set as intensity increases

Asymptotics of defect area, perimeter, and vertices are derived

The nucleus converges to the Steiner point of the convex body

Abstract

Consider a planar random point process made of the union of a point (the origin) and of a Poisson point process with a uniform intensity outside a deterministic set surrounding the origin. When the intensity goes to infinity, we show that the Voronoi cell associated with the origin converges from above to a deterministic convex set. We describe this set and give the asymptotics of the expectation of its defect area, defect perimeter and number of vertices. On the way, two intermediary questions are treated. First, we describe the mean characteristics of the Poisson-Voronoi cell conditioned on containing a fixed convex body around the origin and secondly, we show that the nucleus of such cell converges to the Steiner point of the convex body. As in R\'enyi and Sulanke's seminal papers on random convex hulls, the regularity of the convex body has crucial importance. We deal with both the smooth and polygonal cases. Techniques are based notably on accurate estimates of the area of the Voronoi flower and of the support function of the cell containing the origin as well as on an Efron-type relation.