Abstract voronoi diagram in 3-space

We propose a class of abstract Voronoi diagrams in 3-space that generalizes the planar abstract Voronoi diagram of Klein. Our class of abstract Voronoi diagrams includes the Voronoi diagram of point sites in general position under any convex distance function. To characterize the abstract Voronoi diagram in 3-space we introduce the notion of intersection characteristic. We determine the intersection characteristic for the simplex, the L_∞, and the L_p distance function. We find that the intersection characteristic in case of the simplex distance function is similar to that of the usual Euclidean distance. This enables us to give a randomized incremental algorithm for computing the Voronoi diagram under the simplex distance function in quadratic expected time