The Typical Cell of a Voronoi Tessellation on the Sphere

Abstract

AbstractThe typical cell of a Voronoi tessellation generated by $$n+1$$ n + 1 uniformly distributed random points on the d-dimensional unit sphere $$\mathbb {S}^d$$ 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 $$\mathbb {R}^d$$ R d . Explicit formulas for the expected f-vector are provided for any d and the low-dimensional cases $$d\in \{2,3,4\}$$ d ∈ { 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.