Inscribing cubes and covering by rhombic dodecahedra via equivariant topology

TL;DR

This paper uses equivariant topology to prove that symmetric convex bodies in three dimensions can inscribe cubes and that sets of diameter 1 can be covered by rhombic dodecahedra, revealing new geometric connections.

Contribution

It establishes a new topological approach to inscribing cubes and covering sets with rhombic dodecahedra, linking these problems through equivariant topology.

Findings

Every symmetric convex body in R^3 admits an inscribed cube.

Sets of diameter 1 in R^3 can be covered by rhombic dodecahedra.

Connections between inscribing and covering problems are revealed.

Abstract

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).)