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