Computing closest and farthest points for a query segment

Computing Closest Points for Segments

Farthest line segment Voronoi diagrams

The farthest line segment Voronoi diagram shows properties different from both the closest-segment Voronoi diagram and the farthest-point Voronoi diagram. Surprisingly, this structure did not receive attention in the computational geometry literature. We analyze its combinatorial and topological properties and outline an O(n log n) time construction algorithm that is easy to implement. No restr...

The Mazur Intersection Property and Farthest Points

K. S. Lau had shown that a reflexive Banach space has the Mazur Intersection Property (MIP) if and only if every closed bounded convex set is the closed convex hull of its farthest points. In this work, we show that in general this latter property is equivalent to a property stronger than the MIP. As corollaries, we recapture the result of Lau and characterize the w*-MIP in dual of RNP spaces.

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 ...

On the Farthest Line-Segment Voronoi Diagram

The farthest line-segment Voronoi diagram shows properties surprisingly different than the farthest point Voronoi diagram: Voronoi regions may be disconnected and they are not characterized by convexhull properties. In this paper we introduce the farthest line-segment hull, a cyclic structure that relates to the farthest line-segment Voronoi diagram similarly to the way an ordinary convex hull ...