Extremes

, Volume 17, Issue 3, pp 359–385 | Cite as

Extreme values for characteristic radii of a Poisson-Voronoi Tessellation

Article

Abstract

A homogeneous Poisson-Voronoi tessellation of intensity γ 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.

Keywords

Voronoi tessellations Poisson point process Random covering of the sphere Extremes Boundary effects 

AMS 2010 Subject Classifications:

60D05 62G32 60F05 52A22 

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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Université de RouenLMRS, avenue de l’UniversitéSaint-Etienne-du-Rouvray cedexFrance

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