Inscribing cubes and covering by rhombic dodecahedra via equivariant topology

Details Inscribing cubes and covering by rhombic dodecahedra via equivariant topology Inscribing cubes and covering by rhombic dodecahedra via equivariant topology

1999-06-10

arxiv.org/abs/math/9906066v2

Mathematics

Metric Geometry

a new section added with the proof of a new theorem; 31 pages; to appear in Mathematika

Scientific paper

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).)

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