The typical cell of a Voronoi tessellation on the sphere

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 $din{2,3,4}$ are studied separately. This implies an explicit formula for the total number of $k$-dimensional faces in the spherical Voronoi tessellation as well.