A Lower Bound on Voronoi Diagram Complexity # Boris Aronov + Summary: A Lower Bound on Voronoi Diagram Complexity # Boris Aronov + November 20, 2001 The Voronoi diagram is a much­studied object in geometry. Informally, one starts with a set of disjoint sites, say, points in the plane, and a measure of distance, say, the Euclidean metric. The Voronoi diagram is then a classification of points of the ambient space according to the identity of the closest site or sites. In the above example, it partitions the plane into n open convex Voronoi cells each consisting of points that are strictly closer to a specific site than to any other one, Voronoi edges consisting of points for which exactly two sites are simultaneously closest, and Voronoi vertices which are points from which three or more sites are closest. The number of cells, edges, and vertices is the combinatorial complexity of the diagram. In d dimensions, the diagram is defined analogously and in general contains faces of all dimensions, from 0 up to d. Its complexity is the total number of faces of all dimensions. The Voronoi diagram can similarly be defined in an arbitrary space (such as R d , for any d > 0, d­dimensional sphere S d , etc.) and for any ``reasonable'' metric, such as an L p metric, the metric defined by a convex unit ball, or even some ``distance functions'' which need not be metrics. One can also allow di#erent types of sites---single points, flats, convex bodies, or more general objects. Interested reader may consult the references [4, 8, 1, 6, 10, 7] for di#erent variants of Voronoi diagrams that have been considered in the literature. With very few exceptions, the Voronoi diagram of n pairwise disjoint (possibly non­point) sites Collections: Computer Technologies and Information Sciences