Three-dimensional antipodal and norm-equilateral sets

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.