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.

43 pages, 4 figures, New Trends in Intuitive Geometry, Springer, to appear

References in corpus (18)

• The geometry of Minkowski spaces -- a survey. Part I
• Unit distances and diameters in Euclidean spaces
• Semidefinite programming, multivariate orthogonal polynomials, and codes in spherical caps
• A lower bound for the equilateral number of normed spaces
• A survey on the kissing numbers
• Equilateral sets in finite-dimensional normed spaces
• The geometry of homothetic covering and illumination
• The local Steiner problem in finite-dimensional normed spaces
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• Pairwise intersecting homothets of a convex body
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• Examples of infinite dimensional Banach spaces without infinite equilateral sets
• Vertex degrees of Steiner Minimal Trees in ${\ell }_p^d$ and other smooth Minkowski spaces
• Ball packings with high chromatic numbers from strongly regular graphs
• Sphere-of-influence graphs in normed spaces
• Contact numbers for sphere packings

Cited by in corpus (6)

• Pairwise intersecting homothets of a convex body
• A note on Borsuk's problem in Minkowski spaces
• On the maximum size packings of disks with kissing radius 3
• General penny graphs are at most 43/18-dense
• Few distance sets in ${\ell }_p$ spaces and ${\ell }_p$ product spaces
• Antipodal Hadwiger numbers of finite-dimensional Banach spaces