Recent Developments and Open Problems in Voronoi Diagrams

This paper deals with some open problems in computational geometry related to Voronoi diagrams. The Voronoi diagram and Dealunay triangulation of n points in Ropf^d has complexity Theta (n ^[d/2]) in the worst case. However, the complexity is not high if points are random. Statistical properties of Voronoi diagrams of random points have been studied for decades. The expected complexity of the Voronoi diagram of n random points in the three-dimensional cube is O(n). If n points are generated uniformly at random in the unit ball in R^d, the Voronoi diagram has expected complexity d^O(d)n.