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Combinatorial Distance Geometry in Normed Spaces

  • Konrad J. SwanepoelEmail author
Chapter
Part of the Bolyai Society Mathematical Studies book series (BSMS, volume 27)

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.

Notes

Acknowledgements

We thank Tomasz Kobos, István Talata and a very thorough anonymous referee for providing corrections to a previous version.

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© János Bolyai Mathematical Society and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsLondon School of Economics and Political ScienceLondonUnited Kingdom

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