Voronoi Cells of Varieties with respect to Wasserstein Distances

Abstract: Voronoi diagrams are partitions of a metric space into Voronoi cells according to distance from points on some set w.r.t. some distance. In this thesis we examine Voronoi diagrams of manifolds and varieties w.r.t. the Wasserstein distance from probability theory. We give some upper and lower bounds on the dimension of Voronoi cells based on the geometry of the manifolds and Wasserstein distance balls. We provide an upper bound on the number of full-dimensional Voronoi cells of algebraic varieties and show examples of the bound being tight.