Inscribing cubes and covering by rhombic dodecahedra via equivariant topology

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.