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...

In this paper, we first present some sufficient conditions for the upper semicontinuity and/or the continuity of the Bregman farthest-point map QgC and the relative farthest-point map S g C for a nonempty D-maximally approximately compact subset C of a Banach space X. We next present certain sufficient conditions as well as equivalent conditions for a Klee set to be singleton in a Banach space ...

Given a set S of n static points and a free point p in the Euclidean plane, we study a new variation of the minimum enclosing circle problem, in which a dynamic weight that equals to the reciprocal of the distance from the free point p to the undetermined circle center is included. In this work, we prove the optimal solution of the new problem is unique and lies on the boundary of the farthest-...

The concept of convex polygon-offset distance function was introduced in 2001 by Barequet, Dickerson, and Goodrich. Using this notion of point-to-point distance, they showed how to compute the corresponding nearestand farthest-site Voronoi diagram for a set of points. In this paper we generalize the polygon-offset distance function to be from a point to any convex object with respect to an m-si...

We study the farthest-point distance function, which measures the distance from z ∈ C to the farthest point or points of a given compact set E in the plane. The logarithm of this distance is subharmonic as a function of z, and equals the logarithmic potential of a unique probability measure with unbounded support. This measure σE has many interesting properties that reflect the topology and geo...

We describe a fast sampling algorithm for generating uniformly-distributed point patterns with good blue noise characteristics. The method, based on constrained farthest point optimization, is provably optimal and may be easily parallelized, resulting in an algorithm whose performance/quality tradeoff is superior to other state-of-theart approaches.

Consider the continuum of points along the edges of a network, i.e., a connected, undirected graph with positive edge weights. We measure the distance between these points in terms of the weighted shortest path distance, called the network distance. Within this metric space, we study farthest points and farthest distances. We introduce a data structure supporting queries for the farthest distan...

Given a set S of n static points and a mobile point p in R, we study the variations of the smallest circle that encloses S ∪ {p} when p moves along a straight line `. In this work, a complete characterization of the locus of the center of the minimum enclosing circle (MEC) of S ∪{p}, for p ∈ `, is presented. The locus is a continuous and piecewise differentiable linear function, and each of its...