  In Euclidean space, the asymptotic shape of large cells in various types of
Poisson driven random tessellations has been the subject of a famous conjecture
due to David Kendall. Since shape is a geometric concept and large cells are
identified by means of geometric size functionals, the resolution of the
conjecture is inevitably connected with geometric inequalities of isoperimetric
type and their improvements in the form of geometric stability results,
relating geometric size functionals and hitting functionals. The latter are
deterministic characteristics of the underlying random tessellation. The
current work explores specific and typical cells of random tessellations in
spherical space. A key ingredient of our approach are new geometric
inequalities and quantitative strengthenings in terms of stability results for
quite general and some specific size and hitting functionals of spherically
convex bodies. As a consequence we obtain probabilistic deviation inequalities
and asymptotic distributions of quite general size functionals. In contrast to
the Euclidean setting, where the asymptotic regime concerns large size, in the
spherical framework the asymptotic analysis is concerned with high intensities.

Probability;Metric Geometry