# Combinatorial distance geometry in normed spaces

## TL;DR

This paper surveys combinatorial geometry problems involving distances in normed spaces, discussing properties of various graphs, sets, and measures, and introduces new results on convex bodies, thin cones, and related combinatorial problems.

## Contribution

It provides a comprehensive survey of distance-related problems in normed spaces and presents new results on Hadwiger numbers, blocking numbers, and applications of thin cones.

## Key findings

- Use of Brass's angular measure to prove Hadwiger and blocking number properties.
- New results on Hadwiger numbers and convex bodies in the plane.
- Application of thin cones to distinct distances and combinatorial problems.

## 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.