A random cell splitting scheme on the sphere

TL;DR

This paper introduces a spherical analogue of the STIT tessellation process, analyzing its properties and comparing them to Euclidean cases to understand the influence of curvature on stochastic tessellations.

Contribution

It develops a novel random cell splitting scheme on the sphere and computes key geometric and probabilistic properties using martingale and integral geometry methods.

Findings

Computed first-order moments for various tessellation parameters

Derived the capacity functional for the spherical tessellation

Characterized the point process from intersections with a fixed great circle

Abstract

A random recursive cell splitting scheme of the $2$-dimensional unit sphere is considered, which is the spherical analogue of the STIT tessellation process from Euclidean stochastic geometry. First-order moments are computed for a large array of combinatorial and metric parameters of the induced splitting tessellations by means of martingale methods combined with tools from spherical integral geometry. The findings are compared with those in the Euclidean case, making thereby transparent the influence of the curvature of the underlying space. Moreover, the capacity functional is computed and the point process that arises from the intersection of a splitting tessellation with a fixed great circle is characterized.