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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: [email protected]
E. Makai Jr.
Affiliation:
Alfréd Rényi Mathematical Institute, Hungarian Academy of Sciences, H-1364 Budapest, Pf. 127. Hungary. E-mail: [email protected]
A. Szűcs
Affiliation:
L. Eötvös University, Department of Analysis, H-1088 Múzeum krt. 6–8, Budapest, Hungary. E-mail: [email protected]
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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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References

[AiZi]Aigner, M. and Ziegler, G. M.. Proofs from THE BOOK, 2nd corrected printing (Springer-Verlag, Berlin, 1999). MR 2000k:00001.Google Scholar
[BaBo]Babenko, I. K. and Bogatyi, S. A.. On mapping a sphere into a Euclidean space (Russian). Mat. Zam. 46(3), (1989), 3–8, MR 90m:55003.Google Scholar
[BoKh]Bogatyi, S. A. and Khimshiashvili, G. N.. A generalization of the Borsuk-Ulam theorem and a problem of Knaster (Russian). Soohshch. Akad. Nauk. Gruzinskoj SSR, 123 (1986), 477–480, MR 88h:55004.Google Scholar
[BoMaSo]Boltyanski, V., Martini, H. and Soltan, P. S.. Excursions into Combinatorial Geometry (Springer-Verlag, Berlin, 1997), MR 98b:52001.CrossRefGoogle Scholar
[BoFe]Bonnesen, Th. and Fenchel, W.. Theorie der konvexen Körper (Springer-Verlag, Berlin, 1934): Berichtigter Reprint, Springer-Verlag, Berlin-New York, 1974, Zbl. 8,77,08; MR 49#9736.Google Scholar
[Bor]Borsuk, K.. Drei Satze iiber die n-dimensionale euklidische Sphäre. Fund. Math., 20 (1933), 177–190, Zbl. 6,424,03CrossRefGoogle Scholar
[BroJan]Brocker, Th. and Jänich, K.. Introduction to Differential Topology (Cambridge Univ. Press, Cambridge-New York, 1982). MR831:58001.Google Scholar
[Chen]Chen, W.. Counterexamples to Knaster's conjecture. Topology, 37(2), (1998), 401–405. MR99c:55002.CrossRefGoogle Scholar
[DuKhSh]Duncan, J., Khavinson, D. and Shapiro, H.. Rectangular parallelepipeds in ellipsoids. SIAM Review 38 (1996), 655–657, MR 97f:51039.CrossRefGoogle Scholar
[Dyson]Dyson, F. J.. Continuous functions defined on spheres. Ann. of Math. 54 (1951), 534–536, MR 13–450.CrossRefGoogle Scholar
[Eggll]Eggleston, H. G.. Figures inscribed in convex sets. Amer. Math. Monthly 65 (1958), 76–80, MR 20#4235.CrossRefGoogle Scholar
[Eggl2]Eggleston, H. G.. Covering a three-dimensional set with sets of smaller diameter. J. London Math. Soc. 30 (1955), 11–24, MR 16–743.CrossRefGoogle Scholar
[Floyd]Floyd, E. E.. Real valued mappings of spheres. Proc. Amer. Math. Soc. 6 (1955), 957–959, MR 17–518.CrossRefGoogle Scholar
[Gale]Gale, D.. On inscribing n-dimensional sets in a regular n-simplex. Proc. Amer. Math. Soc. 4 (1953), 222–225, MR 14–787.Google Scholar
[Griff]Griffiths, H. B.. The topology of square pegs in round holes. Proc. London Math. Soc. (3), 62 (1991), 647–672, MR 92h:55004.CrossRefGoogle Scholar
[Gro]Gromov, M. L.. Simplices inscribed to a hypersurface (Russian). Mat. Zam. 5 (1969), 81–89, MR 39#6220.Google Scholar
[GAP]Schönert, M.et al. GAP—Groups, Algorithms and Programming. Lehrstuhl D für Math., Rheinisch-Westfälische Hochschule, Aachen, 5th ed. (1995),.Google Scholar
[Grtin]Grünbaum, B.. A simple proof of Borsuk's conjecture in three dimensions. Proc. Cambr. Phil. Soc. 53 (1957), 776–778, MR 19–763.CrossRefGoogle Scholar
[HMSz]Hausel, T., Makai, E. Jr. and Szűcs, A., Polyhedra inscribed and circumscribed to convex bodies. General Mathematics 5 (1997), Proc. of 3rd Internat. Workshop on Diff. Geom. and its Applies, and the 1st German Romanian Seminar on Geometry, 1997, Sibiu, Romania, 183–190. MR 2000g:53002.Google Scholar
[Hepp]Heppes, A.. On the partitioning of three-dimensional point sets into sets of smaller diameter (Hungarian). Magyar Tud. Akad. Mat. Fiz. Oszt. Közl. 7 (1957), 413–416, MR20#1952.Google Scholar
[Hir]Hirsch, M. W.. Differential Topology (Springer, New York, Heidelberg, Berlin, 1976), MR 56#6669.CrossRefGoogle Scholar
[Hopf]Hopf, H.. Verallgemeinerung bekannter Abbildungs- und Uberdeckungssätze. Portugal. Math. 4 (1944), 129–139, MR 6–165.Google Scholar
[How]Howard, R.. The kinematic formula in Riemannian homogeneous spaces. Memoirs of the Amer. Math. Soc., 106 (1993), no. 509, Amer. Math. Soc, Providence, R.I., MR 94d:53114.CrossRefGoogle Scholar
[Jer]Jerrard, R. P.. Inscribed squares in plane curves. Trans. Amer. Math. Soc., 98 (1961), 234–241, MR 22#11354.CrossRefGoogle Scholar
[KhKa]Kahn, J. and Kalai, G.. A counterexample to Borsuk's conjecture. Bull. Amer. Math. Soc., 29 (1993), 60–62, MR 94a:52007.CrossRefGoogle Scholar
[KIW]Klee, V. and Wagon, S.. Old and New Unsolved Problems in Plane Geometry and Number Theory (Mathematical Association of America, 1991). MR 92k:00014.CrossRefGoogle Scholar
[Knas]Knaster, B., Problème 4. Colloquium Math. 1 (1948), 30–31.Google Scholar
[Kup]Kuperberg, G.. Circumscribing constant-width bodies with polytopes. New York J. Math., 5 (1999), 91–100, arXiv:math.MG/9809165. MR 2OOOh:52OO2.Google Scholar
[Las]Lassak, M.. Private communication (1992),.Google Scholar
[Liv]Livesay, G. R.. On a theorem of F. J. Dyson. Ann. of Math., 59 (1954), 227–229, MR 15–548.CrossRefGoogle Scholar
[MakO]Makeev, V. V.. Dimension restrictions in problems of combinatorial geometry (Russian). Sibirsk. Mat. Zh. 23 (1982), (4), 197–201, 222, MR83m:52019.Google Scholar
[Makl]Makeev, V. V.. Universal covers II (Russian). Ukr. Geom. Sb., 25 (1982), 82–86, MR 84d:52005.Google Scholar
[Mak2]Makeev, V. V.. Some properties of continuous mappings of spheres and problems in combinatorial geometry (Russian). Geometrical Questions in the Theory of Functions and Sets, Kalinin, (1986), 75–85, MR 9Oh:00005.Google Scholar
[Mak3]Makeev, V. V.. The Knaster problem on the continuous mappings from a sphere to a Euclidean space. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 167 (1988), Issled. Topol. 6, 169–178, 193 (Russian, English summary) = J. Soviet Math., 52(1), (1990), MR 90e:55OO6.Google Scholar
[Mak4]Makeev, V. V., Inscribed and circumscribed polyhedra to a convex body (Russian). Mat. Zametki, 55(4), (1994), 128–130 = Math. Notes, 55 (1994), 423–425, MR 95h:52002.Google Scholar
[Mak5]Makeev, V. V.. On affine images of a rhombo-dodecahedron circumscribed about a three-dimensional convex body (Russian). Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMl) 246 (1997), Geom. i Topol. 2, 191–195, 200. MR 99e:52005.Google Scholar
[Mak6]Makeev, V. V.. Some special configurations of planes that are associated with convex compacta. Zap. Nauchn. Sem. S. Peterburg. Otdel. Mat. Inst. Steklov. 252 (1998), Geom. i Topol. 3, 165–174, 251. MR 2001f:52013.Google Scholar
[MiSt]Milnor, J. W. and Stasheff, J. D.. Characteristic Classes (Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo 1974), MR 55#13428.Google Scholar
[Pal]Pal, J.. Über ein elementares Variationsproblem. Kgl. Danske Vid. Selskab., Mat-Fys. Medd., 3 (1920), 3–35, Jahrbuch Fortschr. Math., 47 (1919–20), 684.Google Scholar
[Steen]Steenrod, N.. Topology of Fibre Bundles (Princeton Mathematical Series, vol. 14, Princeton University Press, Princeton, N.J., 1951), MR 12–522.CrossRefGoogle Scholar
[YaYu]Yamabe, H. and Yujobô, Z.. On the continuous function defined on a sphere. Osaka Math. J., 2 (1950), 19–22, MR 12–198.Google Scholar
[Yang]Yang, C. T.. On theorems of Borsuk-Ulam, Kakutani-Yamabe-Yujobo and Dyson, 1. Ann. of Math. (2), 60 (1954), 262–282, MR 16–502d.CrossRefGoogle Scholar