Inscribing cubes and covering by rhombic dodecahedra via equivariant topology
| dc.creator | Hausel, Tamas | |
| dc.creator | Makai Jr., Endre | |
| dc.creator | Szucs, Andras | |
| dc.date | 1999-06-10 | |
| dc.date | 2000-04-20 | |
| dc.date.accessioned | 2026-07-07T05:29:26Z | |
| dc.date.available | 2026-07-07T05:29:26Z | |
| dc.description | First, we prove a special case of Knaster's problem, implying that each symmetric convex body in R^3 admits an inscribed cube. We deduce it from a theorem in equivariant topology, which says that there is no S_4-equivariant map from SO(3) to S^2, where S_4 acts on SO(3) as the rotation group of the cube and on S^2 as the symmetry group of the regular tetrahedron. We also give some generalizations. Second, we show how the above non-existence theorem yields Makeev's conjecture in R^3 that each set in R^3 of diameter 1 can be covered by a rhombic dodecahedron, which has distance 1 between its opposite faces. This reveals an unexpected connection between inscribing cubes into symmetric bodies and covering sets by rhombic dodecahedra. Finally, we point out a possible application of our second theorem to the Borsuk problem in R^3. (Similar results were obtained recently by V.V. Makeev and independently by G. Kuperberg (cf. math.MG/9809165).) | |
| dc.description | a new section added with the proof of a new theorem; 31 pages; to appear in Mathematika | |
| dc.identifier | https://arxiv.org/abs/math/9906066 | |
| dc.identifier | http://arxiv.org/abs/math/9906066 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/78638 | |
| dc.subject | Metric Geometry | |
| dc.subject | Geometric Topology | |
| dc.subject | 52A15, 55Mxx | |
| dc.title | Inscribing cubes and covering by rhombic dodecahedra via equivariant topology | |
| dc.type | text |