Extreme values for characteristic radii of a Poisson-Voronoi tessellation

TL;DR

This paper analyzes the extreme values of inradius and circumscribed radius in a Poisson-Voronoi tessellation, showing their distributions converge to Gumbel or Weibull types as intensity grows, with implications for tessellation regularity.

Contribution

It provides the first asymptotic distribution results for the extreme characteristic radii of cells in a Poisson-Voronoi tessellation, including boundary effects.

Findings

Maximum and minimum radii converge to Gumbel or Weibull distributions.

Boundary cell contributions are negligible in the asymptotic regime.

Results imply convergence to simplex shape for the smallest circumscribed radius cell.

Abstract

A homogeneous Poisson-Voronoi tessellation of intensity $\gamma$ is observed in a convex body $W$. We associate to each cell of the tessellation two characteristic radii: the inradius, i.e. the radius of the largest ball centered at the nucleus and included in the cell, and the circumscribed radius, i.e. the radius of the smallest ball centered at the nucleus and containing the cell. We investigate the maximum and minimum of these two radii over all cells with nucleus in $W$. We prove that when $\gamma\rightarrow\infty$, these four quantities converge to Gumbel or Weibull distributions up to a rescaling. Moreover, the contribution of boundary cells is shown to be negligible. Such approach is motivated by the analysis of the global regularity of the tessellation. In particular, consequences of our study include the convergence to the simplex shape of the cell with smallest circumscribed radius and an upper-bound for the Hausdorff distance between $W$ and its so-called Poisson-Voronoi approximation.