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Inscribing cubes and covering by rhombic dodecahedra via equivariant topology

Part of: General convexity

Published online by Cambridge University Press:  26 February 2010

T. Hausel ,
E. Makai Jr.  and
A. Szűcs
T. Hausel
Affiliation:
Department of Mathematics, University of California at Berkeley, Berkeley, CA 94720-3840, USA. E-mail: hausel@math.berkeley.edu
E. Makai Jr.
Affiliation:
Alfréd Rényi Mathematical Institute, Hungarian Academy of Sciences, H-1364 Budapest, Pf. 127. Hungary. E-mail: makai@renyi.hu
A. Szűcs
Affiliation:
L. Eötvös University, Department of Analysis, H-1088 Múzeum krt. 6–8, Budapest, Hungary. E-mail: szucs@cs.elte.hu
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Abstract

First, a special case of Knaster's problem is proved implying that each symmetric convex body in ℝ3 admits an inscribed cube. It is deduced 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) on the right as the rotation group of the cube, and on S2 on the right as the symmetry group of the regular tetrahedron. Some generalizations are also given.

MSC classification

Secondary: 52A15: Convex sets in $3$ dimensions (including convex surfaces)
Type
Research Article
Information
Mathematika , Volume 47 , Issue 1-2 , December 2000 , pp. 371 - 397
Copyright
Copyright © University College London 2000

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