The typical cell of a Voronoi tessellation on the sphere

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.