Counting Cells of Order-k Voronoi Tessellations in ℝ³ with Morse Theory

Abstract

Generalizing Lee’s inductive argument for counting the cells of higher order Voronoi tessellations in R2 to R3, we get precise relations in terms of Morse theoretic quantities for piecewise constant functions on planar arrangements. Specifically, we prove that for a generic set of n ≥ 5 points in R3, the number of regions in the order-k Voronoi tessellation is Nk−1 − ( k 2 ) n + n, for 1 ≤ k ≤ n − 1, in which Nk−1 is the sum of Euler characteristics of these function’s first k − 1 sublevel sets. We get similar expressions for the vertices, edges, and polygons of the order-k Voronoi tessellation. 2012 ACM Subject Classification Theory of computation → Computational geometry