Inscribing cubes and covering by rhombic dodecahedra via equivariant topology

First, we prove a special case of Knaster’s problem, implying that each symmetric convex body in R3 admits an inscribed cube. We deduce it 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) 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 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, we point out a possible application of our second theorem to the Borsuk problem in R3.