Title: Combinatorial distance geometry in normed spaces,
Abstract: 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.