Cardinality of simplexes in a Delaunay tessellation

Abstract

Given a set of n linearly independent points in a Euclidean space Ed, P = {p1, ... , pn} with n >; d, a Delaunay tessellation with at lease one d-dimensional simplex can be constructed. This tessellation is unique up to degenerate linearity conditions. The cardinality of the set of unique k-dimensional simplexes, k ≤ d, in the tessellation is bounded and the bound can be computed, given the dimension of the space, d, and the number of points in the tessellation generating set, n. These bounds can be refined if the number of points on the convex hull of P, m, is known. The bounds on cardinality are developed using constructive geometric arguments presented in the sequence necessary to construct the tessellation. The cardinality of simplexes in the Voronoi diagram is then related to the Delaunay tessellation by geometric duality. An example is given.