Euclidean proximity and power diagrams

Power diagrams [Aur87] are a useful generalization of Voronoi diagrams in which the sites defining the diagram are not points but balls. They derive their name from the fact that the distance used in their definition is not the standard Euclidean distance, but instead the classical notion of the power of a point with respect to a ball [Cox61]. The advantage of the power distance is that the ‘bisector surfaces’, the surfaces equidistant from two given balls, are hyper-planes, and thus the Euclidean structure of these diagrams is polyhedral and they are easier to compute and manipulate than standard Voronoi diagrams. Several algorithms and applications for the power diagrams are already known, including those referenced in [IIM85, Aur88]. Most recently power diagrams have played a key role in work related to alpha shapes and ‘skins’ [Ede93, EM94, Ede95], as collections of balls are a natural model for molecular structures. The Voronoi diagram of point sites in any dimension has many useful proximity properties. Let us say that two sites are neighbors in the Voronoi diagram if their Voronoi cells share a common facet. It is well known that, for a set of points S, the closest pair of points in S are neighbors in the Voronoi diagram of S — equivalently, the closest pair of points must define an edge of the dual Delaunay diagram. Even more is true: the nearest neighbor to any point p ∈ S is a neighbor to p in the Voronoi diagram (equivalently, is connected to this neighbor by a Delaunay edge). Surprisingly, it seems that analogous proximity properties have not been studied for the power diagram. In this note, we consider the the power diagram of a set of disjoint balls. We show that the closest pair of