ar X iv : m at h / 05 06 24 0 v 1 [ m at h . M G ] 1 3 Ju n 20 05 THREE - DIMENSIONAL ANTIPODAL AND NORM - EQUILATERAL SETS
نویسنده
چکیده
We characterize the three-dimensional spaces admitting at least six or at least seven equidistant points. In particular, we show the existence of C∞ norms on R 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 R. On the other hand, no differentiable norm on R 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.
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