Voronoi diagrams and Morse theory of the distance function

Abstract

We consider the (minimal) distance function of a point in the plane to a set P of N points in the plane. The locus of non-dierentiability of this distance function consists (besides of the points of P) exactly of the Voronoi diagram of P. We show that the number of minima (m), maxima (M) and `saddle points' (s) of the distance function satisfy: m - s + M = 1. This is similar to the Morse type of statements for dierentiable functions. The saddle points occur exactly where a Delaunay edge cuts the corresponding Voronoi edge in its interior. The set of those edges form a subgraph of the Delaunay graph, which connects all minima and saddle points. This graph devides the plane into regions. In each of the compact regions, there is exactly one maximum, the non compact regions don't contain a local maximum. At the end we classify all those graphs if P contains of 3 or 4 points.