Three-dimensional antipodal and norm-equilateral sets
| dc.creator | Schuermann, Achill | |
| dc.creator | Swanepoel, Konrad | |
| dc.date | 2005-06-13 | |
| dc.date.accessioned | 2026-07-07T07:48:45Z | |
| dc.date.available | 2026-07-07T07:48:45Z | |
| dc.description | 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. | |
| dc.description | 20 pages, 15 figures | |
| dc.identifier | https://arxiv.org/abs/math/0506240 | |
| dc.identifier | http://arxiv.org/abs/math/0506240 | |
| dc.identifier | Pacific Journal of Mathematics 228 (2006), 349--370. | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/124609 | |
| dc.subject | Metric Geometry | |
| dc.subject | Differential Geometry | |
| dc.subject | 52A21 (Primary) 49Q15 (Secondary) | |
| dc.title | Three-dimensional antipodal and norm-equilateral sets | |
| dc.type | text |