Geometric inequalities, stability results and Kendall's problem in spherical space

TL;DR

This paper investigates geometric inequalities and stability results for random tessellations in spherical space, providing new inequalities, probabilistic deviation bounds, and asymptotic distribution results, contrasting with Euclidean space where large size is the focus.

Contribution

It introduces novel geometric inequalities and stability results for spherical convex bodies, extending the understanding of random tessellations in spherical geometry.

Findings

New geometric inequalities for spherical convex bodies.

Probabilistic deviation inequalities for size functionals.

Asymptotic distributions at high intensities in spherical space.

Abstract

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.