Discrete & Computational Geometry

Let S be a finite collection of compact convex sets in \R ^d . Let D(S) be the largest diameter of any member of S . We say that the collection S is -separated if, for every 0 < k < d , any k of the sets can be separated from any ...

We prove that the red--blue discrepancy of the set system formed by n points and n axis-parallel boxes in <bo>R</bo> ^d can be as high as n ^Ω(1) in any dimension d= Ω(log n) . This contrasts with the fixed-dimensional case d=O(1) , where the ...

This paper describes an algorithm for maintaining an approximating triangulation of a deforming surface in R ³ . The surface is the envelope of an infinite family of spheres defined and controlled by a finite collection of weighted points. The ...

Random sampling is an efficient method to deal with constrained optimization problems in computational geometry. In a first step, one finds the optimal solution subject to a random subset of the constraints; in many cases, the expected number of ...

We describe a kinetic data structure (KDS) that maintains the connected components of the union of a set of unit-radius disks moving in the plane. We assume that the motion of each disk can be specified by a low-degree algebraic trajectory; this ...

Given a set of n black and n white points in general position in the plane, a line l determined by them is said to be balanced if each open half-plane bounded by l contains precisely the same number of black points as white points. It is proved that the ...

It is shown that every set of n points in the plane has an element from which there are at least cn ^6/7 other elements at distinct distances, where c>0 is a constant. This improves earlier results of Erd s, Moser, Beck, Chung, Szemerédi, ...