  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).)
