An earlier result states that a point of the surface of a convex polyhedron with n vertices, endowed with its intrinsic metric, cannot have more than n antipodes (farthest points). In this paper we produce examples of polyhedra with n vertices, on which some suitable point admits exactly n antipodes. MSC (2000): 52B10, 53C45.

In this note, we outline a very simple algorithm for the following problem: Given a set S of n points p1, p2, p3, . . . , pn in the plane, we have O(n ) segments implicitly defined on pairs of these n points. For each point pi, find a segment from this set of implicitly defined segments that is farthest from pi. The complexity of our algorithm is in O(nh+n logn), where n is the number of input ...

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 ...

We present structural properties of the farthest line-segment Voronoi diagram in the piecewise linear L∞ and L1 metrics, which are computationally simpler than the standard Euclidean distance and very well suited for VLSI applications. We introduce the farthest line-segment hull, a closed polygonal curve that characterizes the regions of the farthest line-segment Voronoi diagram, and is related...

Sunto Dopo una breve introduzione delle principali idee della topologia formale [Sam87] vengono introdotte le nozioni di punto e co-punto. Si presentano quindi alcuni metodi per costruire punti e co-punti in opportune topologie formali e si forniscono alcune applicazioni logiche di tali costruzioni 1 .

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...

The Voronoi diagram is a fundamental geometric structure that encodes proximity information. Given a set of geometric objects, called sites, their Voronoi diagram is a subdivision of the underlying space into maximal regions, such that all points within one region have the same nearest site. Problems in diverse application domains (such as VLSI CAD, robotics, facility location, etc.) demand var...