### 2005-19

by Achill Schürmann, Konrad Swanepoel.

**Series:** 2005-19, Preprints

- MSC:
- 52A21 Finite-dimensional Banach spaces (including special norms, zonoids, etc.)
- 49Q15 Geometric measure and integration theory, integral and normal currents

**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.

**Keywords:**

antipodal sets, equidistant sets, energy-minimizing surfaces, convex norms