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Three-dimensional antipodal and norm-equilateral sets

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

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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:
Additional Information: © 2009 Pacific Journal of Mathematics
Divisions: Mathematics
Subjects: Q Science > QA Mathematics
Date Deposited: 16 Oct 2009 09:47
Last Modified: 16 May 2024 00:29

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