  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.
