Bounds on Delaunay tessellations

Abstract

Motivated by applications in density estimation and data compression, this article considers the bounds on the number of tiles in a Delaunay tessellation as a function of both the number of tessellating points and the dimension. Results can also be interpreted for the dual of the Delaunay tessellation, the Voronoi diagram. Several theoretical lower and upper bounds are found in the combinatorics and computational geometry literatures and are brought together in this article. We make a comparison of these bounds with several empirically derived curves based on multivariate uniform and Gaussian generated random tessellating points. The upper bounds are found to be very conservative when compared with the empirically derived number of tiles, often off by many orders of magnitude.