Computing Farthest Neighbors on a Convex Polytope

Let be a set of points in convex position in . The farthest-point Voronoi diagram of partitions into convex cells. We consider the intersection of the diagram with the boundary of the convex hull of . We give an algorithm that computes an implicit representation of in expected time. More precisely, we compute the combinatorial structure of , the coordinates of its vertices, and the equation of the plane defining each edge of . The algorithm allows us to solve the all-pairs farthest neighbor problem for in expected time , and to perform farthest-neighbor queries on in time with high probability.