The B. and M. Shapiro conjecture stated that all solutions of the Schubert Calculus problems associated with real points on the rational normal curve should be real. For Grassmannians, it was prove...

Given a set of moving points in ℝd, we show how to cluster them in advance, using a small number of clusters, so that at any time this static clustering is competitive with the optimal k-center clustering at that time. The advantage of this approach is that it avoids updating the clustering as time passes. We also show how to maintain this static clustering efficiently under insertions and deletions. To implement this static clustering efficiently, we describe a simple technique for speeding up clustering algorithms and apply it to achieve faster clustering algorithms for several problems. In particular, we present a linear time algorithm for computing a 2-approximation to the k-center clustering of a set of n points in ℝd. This slightly improves the algorithm of Feder and Greene, that runs in Θ(n log k) time (which is optimal in the algebraic decision tree model).

We characterize ellipsoids among convex bodies in Ed looking at the sections parallel to two or three hyperplanes.

The “thirteen spheres problem,” also known as the “Gregory–Newton problem,” is to determine the maximum number of three-dimensional spheres that can simultaneously touch a given sphere, where all the spheres have the same radius. The history of the problem goes back to a disagreement between Isaac Newton and David Gregory in 1694. Using a combination of harmonic analysis and linear programming it can be shown that the maximum cannot exceed 13, but in fact 13 is impossible. The standard proof that the maximum is 12 uses an ad hoc construction that does not appear to extend to higher dimensions. In this paper we describe a new proof that uses linear programming bounds and properties of spherical Delaunay triangulations.

We investigate universal bounds on spherical codes and spherical designs that could be obtained using Delsarte’s linear programming methods. We give a lower estimate for the LP upper bound on codes, and an upper estimate for the LP lower bound on designs. Specifically, when the distance of the code is fixed and the dimension goes to infinity, the LP upper bound on codes is at least as large as the average of the best known upper and lower bounds. When the dimension n of the design is fixed, and the strength k goes to infinity, the LP bound on designs turns out, in conjunction with known lower bounds, to be proportional to kn-1.

We introduce for oriented matroids a generalization of the concepts of intersection and linking numbers in Euclidean space, with most of their main properties (see Wu). As an application, we reprove a result of Brehm in a slightly extended form.

In the one-round Voronoi game, the first player chooses an n-point set W in a square Q, and then the second player places another n-point set B into Q. The payoff for the second player is the fraction of the area of Q occupied by the regions of the points of B in the Voronoi diagram of W \cup B. We give a (randomized) strategy for the second player that always guarantees him a payoff of at least ½ + α, for a constant α > 0 and every large enough n. This contrasts with the one-dimensional situation, with Q=[0,1], where the first player can always win more than ½.