Swanepoel, Konrad 
ORCID: 0000-0002-1668-887X and Schurmann, Achill
(2006)
Three-dimensional antipodal and norm-equilateral sets.
Pacific Journal of Mathematics, 228 (2).
pp. 349-370.
ISSN 0030-8730
Abstract
We characterize three-dimensional spaces admitting at least six or at least seven equidistant points. In particular, we show the existence of C∞ norms on ℝ3 admitting six equidistant points, which refutes a conjecture of Lawlor and Morgan (1994, Pacific J. Math. 166, 55–83), and gives the existence of energy-minimizing cones with six regions for certain uniformly convex norms on ℝ3. On the other hand, no differentiable norm on ℝ3 admits seven equidistant points. A crucial ingredient in the proof is a classification of all three-dimensional antipodal sets. We also apply the results to the touching numbers of several three-dimensional convex bodies.
| Item Type: | Article |
|---|---|
| Official URL: | http://pjm.math.berkeley.edu/pjm/2009/242-2/index.... |
| Additional Information: | © 2009 Pacific Journal of Mathematics |
| Divisions: | Mathematics |
| Subjects: | Q Science > QA Mathematics |
| Date Deposited: | 16 Oct 2009 09:47 |
| Last Modified: | 31 Jan 2025 07:00 |
| URI: | http://eprints.lse.ac.uk/id/eprint/25450 |
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