ON THE EXISTENCE OF 1-SEPARATED SEQUENCES ON THE UNIT BALL OF A FINITE-DIMENSIONAL BANACH SPACE

Given a finite-dimensional Banach space X and an Auerbach basis {(xk, xk∗) : 1 ≤ k ≤ n} of X, it is proved that there exist n + 1 linear combinations z1, ... , zn+1 of x1, ... , xn with coordinates 0, ±1, such that ∥zk∥ = 1, for k = 1, 2, ... , n + 1 and ∥zk - zl∥ > 1, for 1 ≤ k < l ≤ n + 1. Copyright © University College London 2014

Combinatorial distance geometry in normed spaces

We survey problems and results from combinatorial geometry in normed spaces, concentrating on problems that involve distances. These include various properties of unit-distance graphs, minimum-distance graphs, diameter graphs, as well as minimum spanning trees and Steiner minimum trees. In particular, we discuss translative kissing (or Hadwiger) numbers, equilateral sets, and the Borsuk problem in normed spaces. We show how to use the angular measure of Peter Brass to prove various statements about Hadwiger and blocking numbers of convex bodies in the plane, including some new results. We also include some new results on thin cones and their application to distinct distances and other combinatorial problems for normed spaces