---
res:
bibo_abstract:
- First, a special case of Knaster's problem is proved implying that each symmetric
convex body in ℝ3 admits an inscribed cube. It is deduced from a theorem in equivariant
topology, which says that there is no S4 - equivariant map from SO(3) to S2, where
S4 acts on SO(3) on the right as the rotation group of the cube, and on S2 on
the right as the symmetry group of the regular tetrahedron. Some generalizations
are also given. Second, it is shown how the above non-existence theorem yields
Makeev's conjecture in ℝ3 that each set in ℝ3 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, a possible application of our
second theorem to the Borsuk problem in ℝ3 is pointed out.@eng
bibo_authorlist:
- foaf_Person:
foaf_givenName: Tamas
foaf_name: Tamas Hausel
foaf_surname: Hausel
foaf_workInfoHomepage: http://www.librecat.org/personId=4A0666D8-F248-11E8-B48F-1D18A9856A87
- foaf_Person:
foaf_givenName: Endre
foaf_name: Makai, Endre
foaf_surname: Makai
- foaf_Person:
foaf_givenName: András
foaf_name: Szücs, András
foaf_surname: Szücs
bibo_doi: 10.1112/S0025579300015965
bibo_issue: 1-2
bibo_volume: 47
dct_date: 2000^xs_gYear
dct_publisher: University College London@
dct_title: Inscribing cubes and covering by rhombic dodecahedra via equivariant
topology@
...