Inscribing cubes and covering by rhombic dodecahedra via equivariant topology
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_4equivariant 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 nonexistence 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).)
 Publication:

arXiv Mathematics eprints
 Pub Date:
 June 1999
 arXiv:
 arXiv:math/9906066
 Bibcode:
 1999math......6066H
 Keywords:

 Mathematics  Metric Geometry;
 Mathematics  Geometric Topology;
 52A15;
 55Mxx
 EPrint:
 a new section added with the proof of a new theorem