The typical cell of a Voronoi tessellation on the sphere

TL;DR

This paper investigates the properties of typical cells in Voronoi tessellations on the sphere, linking their face counts to beta' polytopes and providing explicit formulas for various dimensions.

Contribution

It establishes a distributional equivalence between the typical cell's f-vector and that of beta' polytopes, with explicit formulas for expected face counts in all dimensions.

Findings

Derived explicit formulas for expected f-vectors in any dimension.

Connected spherical Voronoi cells to beta' polytopes in Euclidean space.

Analyzed low-dimensional cases d=2,3,4 separately.

Abstract

The typical cell of a Voronoi tessellation generated by $n+1$ uniformly distributed random points on the $d$-dimensional unit sphere $\mathbb S^d$ is studied. Its $f$-vector is identified in distribution with the $f$-vector of a beta' polytope generated by $n$ random points in $\mathbb R^d$. Explicit formulae for the expected $f$-vector are provided for any $d$ and the low-dimensional cases $d\in\{2,3,4\}$ are studied separately. This implies an explicit formula for the total number of $k$-dimensional faces in the spherical Voronoi tessellation as well.