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A homogeneous Poisson-Voronoi tessellation of intensity $\gamma$ is observedin a convex body $W$. We associate to each cell of the tessellation twocharacteristic radii: the inradius, i.e. the radius of the largest ballcentered 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 thecell. We investigate the maximum and minimum of these two radii over all cellswith nucleus in $W$. We prove that when $\gamma\rightarrow\infty$, these fourquantities converge to Gumbel or Weibull distributions up to a rescaling.Moreover, the contribution of boundary cells is shown to be negligible. Suchapproach is motivated by the analysis of the global regularity of thetessellation. In particular, consequences of our study include the convergenceto the simplex shape of the cell with smallest circumscribed radius and anupper-bound for the Hausdorff distance between $W$ and its so-calledPoisson-Voronoi approximation.

Extreme values for characteristic radii of a Poisson-Voronoi tessellation