Sharp asymptotics for the maximal distance from the boundary to the nucleus of a typical Poisson-Voronoi cell

Pierre Calka; Cecilia d'Errico; Nathanaël Enriquez

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arXiv · 2511.11189

Sharp asymptotics for the maximal distance from the boundary to the nucleus of a typical Poisson-Voronoi cell

Pierre Calka·Cecilia d'Errico·Nathanaël Enriquez

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Abstract

We consider the typical Poisson-Voronoi cell in the Euclidean space R d and in particular the maximal distance D from a vertex of that cell to its nucleus. We provide a sharp asymptotics for the tail distribution of D. As a byproduct, we prove that the extremal index related to the sequence of such distances for all Voronoi cells included in a large box is equal to (2d) -1 . This confirms a conjecture formulated by Chenavier and Robert. The explicit constant appearing in the estimate of the tail probability of D is proved to be the mean volume of a random simplex formed by uniformly distributed points on the unit sphere conditioned on satisfying some spatial condition.

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Pierre Calka, Cecilia d'Errico, Nathanaël Enriquez. 2025-11-14. Sharp asymptotics for the maximal distance from the boundary to the nucleus of a typical Poisson-Voronoi cell. https://arxiv.org/abs/2511.11189

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