Three-dimensional antipodal and norm-equilateral sets

Achill Schürmann and Konrad J. Swanepoel

Vol. 228 (2006), No. 2, 349–370

DOI: 10.2140/pjm.2006.228.349

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.

Keywords

antipodal set, norm-equilateral set, touching number

Mathematical Subject Classification 2000

Primary: 52C17

Secondary: 49Q05, 52A15, 52A21, 52C10

Milestones

Received: 9 March 2005

Revised: 3 June 2005

Accepted: 9 June 2005

Published: 1 December 2006