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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 ... (Update)
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...this does not disprove Conjecture 1. Acknowledgements. The author has learned that Makeev [5] and independently Hausel, Makai, and Szucs [3] have obtained similar results. The author would like to thank Don Chakerian for suggesting the problem and for extensive discussions,...
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BibTeX entry: (Update)
T. Hausel, E. Makai Jr., and A. Szucs, Inscribing cubes and covering by rhombic dodecahedra via equivariant topology , math.MG/9906066. http://citeseer.ist.psu.edu/hausel00inscribing.html More
@misc{ hausel-inscribing,
author = "T. Hausel and E. Jr and A. Szucs",
title = "Inscribing cubes and covering by rhombic dodecahedra via equivariant topology",
text = "T. Hausel, E. Makai Jr., and A. Szucs, Inscribing cubes and covering by
rhombic dodecahedra via equivariant topology , math.MG/9906066.",
url = "citeseer.ist.psu.edu/hausel00inscribing.html" }
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