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# Title: Three-dimensional antipodal and norm-equilateral sets

Abstract: We characterize the three-dimensional spaces admitting at least six or at least seven equidistant points. In particular, we show the existence of $C^\infty$ norms on $\R^3$ admitting six equidistant points, which refutes a conjecture of Lawlor and Morgan (1994, Pacific J. Math \textbf{166}, 55--83), and gives the existence of energy-minimizing cones with six regions for certain uniformly convex norms on $\R^3$. On the other hand, no differentiable norm on $\R^3$ 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.
 Comments: 20 pages, 15 figures Subjects: Metric Geometry (math.MG); Differential Geometry (math.DG) MSC classes: 52A21 (Primary) 49Q15 (Secondary) Journal reference: Pacific Journal of Mathematics 228 (2006), 349--370. Cite as: arXiv:math/0506240 [math.MG] (or arXiv:math/0506240v1 [math.MG] for this version)

## Submission history

From: Achill Schuermann [view email]
[v1] Mon, 13 Jun 2005 12:48:16 GMT (23kb)
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