  We consider the Voronoi tessellation based on a homogeneous Poisson point
process in $\mathbf{R}^{d}$. For a geometric characteristic of the cells (e.g.
the inradius, the circumradius, the volume), we investigate the point process
of the nuclei of the cells with large values. Conditions are obtained for the
convergence in distribution of this point process of exceedances to a
homogeneous compound Poisson point process. We provide a characterization of
the asymptotic cluster size distribution which is based on the Palm version of
the point process of exceedances. This characterization allows us to compute
efficiently the values of the extremal index and the cluster size probabilities
by simulation for various geometric characteristics. The extension to the
Poisson-Delaunay tessellation is also discussed.
