Polyhedral Voronoi diagrams of polyhedra in three dimensions

We show that the complexity of the Voronoi diagram,of a collection of disjoint polyhedra in general position in 3-space that have n vertices overall, under a convex distance function induced by a polyhedron with O(1) facets, is O(n(2+epsilon)), for any epsilon > 0. We also show that when the sites are n segments in 3-space, this complexity is 0(n 2 ot (n) log n). This generalizes previous results by Chew et al. [10] and by Aronov and Sharir [4], and solves an open problem put forward by Agarwal and Shatir [2]. Specific distance functions for which our results hold are the L-1 and L-infinity metrics. These results imply that we can preprocess a collection of polyhedra as above into a near-quadratic data structure that can answer delta-approximate Euclidean nearest-neighbor queries amidst the polyhedra in time O(log(n/delta)), for an arbitrarily small delta > 0.