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Combinatorial Distance Geometry in Normed Spaces

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New Trends in Intuitive Geometry

Part of the book series: Bolyai Society Mathematical Studies ((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.

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Notes

  1. 1.

    Very recently, Serge Vlăduţ [206] found an exponential lower bound for \(H_{L}(B^{d})\).

  2. 2.

    Very recently, Gerencsér and Harangi [80] proved the lower bound \(A'(d)\geqslant 2^{d-1}+1\).

  3. 3.

    Very recently, Vlăduţ showed that \(H_{L}(\mathbb {E}^d) \geqslant 2^{0.0219(n)-0(n)}\), which also gives an exponential lower bound on \(\delta (\mathbb {E}^d)\).

  4. 4.

    Very recently, Aubrey de Grey [53] improved the lower bound to 5.

  5. 5.

    De Grey should that the Euclidean plane is an example.

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Acknowledgements

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

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    Konrad J. Swanepoel

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Swanepoel, K.J. (2018). Combinatorial Distance Geometry in Normed Spaces. In: Ambrus, G., Bárány, I., Böröczky, K., Fejes Tóth, G., Pach, J. (eds) New Trends in Intuitive Geometry. Bolyai Society Mathematical Studies, vol 27. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-57413-3_17

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