Three-dimensional antipodal and norm-equilateral sets

We characterize the three-dimensional spaces admitting at least six or at least seven equidistant points. In particular, we show the existence of $C^{\mathrm{\infty }}$ norms on ${\mathbb{R}}^3$ admitting six equidistant points, which refutes a conjecture of Lawlor and Morgan (1994, Pacific J. Math $\mathbf{\text{166}}$, 55–83), and gives the existence of energy-minimizing cones with six regions for certain uniformly convex norms on ${\mathbb{R}}^3$. On the other hand, no differentiable norm on ${\mathbb{R}}^3$ 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.