Splitting tessellations in spherical spaces

TL;DR

This paper introduces and analyzes splitting tessellations in spherical spaces, deriving statistical properties, correlation functions, and distributions of typical cells, with comparisons to Poisson tessellations.

Contribution

It provides the first comprehensive study of splitting tessellations in spherical spaces, including explicit formulas and distributional results.

Findings

Explicit spherical pair-correlation function computed.

Distributions of typical cells expressed as mixtures of Poisson tessellations.

Expected length and birth time distribution of maximal spherical segments derived.

Abstract

The concept of splitting tessellations and splitting tessellation processes in spherical spaces of dimension $d\geq 2$ is introduced. Expectations, variances and covariances of spherical curvature measures induced by a splitting tessellation are studied using tools from spherical integral geometry. Also the spherical pair-correlation function of the $(d-1)$-dimensional Hausdorff measure is computed explicitly and compared to its analogue for Poisson great hypersphere tessellations. Finally, the typical cell distribution and the distribution of the typical spherical maximal face of any dimension $k\in\{1,\ldots,d-1\}$ are expressed as mixtures of the related distributions of Poisson great hypersphere tessellations. This in turn is used to determine the expected length and the precise birth time distribution of the typical maximal spherical segment of a splitting tessellation.