Poisson–Voronoi Tessellation on a Riemannian Manifold

Abstract In this paper, we consider a Riemannian manifold M and the Poisson–Voronoi tessellation generated by the union of a fixed point x₀ and a Poisson point process of intensity on M. We obtain a two-term asymptotic expansion, when goes to infinity, of the mean number of vertices of the Voronoi cell associated with x₀. The 1st term of the estimate is equal to the mean number of vertices in the Euclidean setting, while the 2nd term involves the scalar curvature of M at x₀. This settles with the proper and rigorous frame the former 2D statement from 19 and extends it to higher dimension. The key tool for proving this result is a new change of variables formula of Blaschke–Petkantschin type in the Riemannian setting, which brings out the Ricci curvatures of the manifold.