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The problem of illumination of the boundary of a convex body by affine subspaces

Part of: General convexity

Published online by Cambridge University Press:  26 February 2010

Károly Bezdek
Károly Bezdek
Affiliation:
Dept. of Geometry, Eötvös L. University, 1088 Budapest, Rákóczi út 5, Hungary
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Abstract

The main result of this paper is the following theorem. If P is a convex polytope of Ed with affine symmetry, then P can be illuminated by eight (d - 3)-dimensional affine subspaces (two (d- 2)-dimensional affine subspaces, resp.) lying outside P, where d ≥ 3. For d = 3 this proves Hadwiger's conjecture for symmetric convex polyhedra namely, it shows that any convex polyhedron with affine symmetry can be covered by eight smaller homothetic polyhedra. The cornerstone of the proof is a general separation method.

MSC classification

Secondary: 52A20: Convex sets in $n$ dimensions (including convex hypersurfaces)
Type
Research Article
Information
Mathematika , Volume 38 , Issue 2 , December 1991 , pp. 362 - 375
Copyright
Copyright © University College London 1991

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References

1.Bezdek, K.. Hadwiger-Levi's covering problem revisited. Progress in Discrete and Computational Geometry (Springer-Verlag). 1–44. To appear.Google Scholar
2.Boltjansky, V. G.. The problem of the illumination of the boundary of a convex body (in Russian), Izvestiya Mold. Fil. Akad. Nauk SSSR, 76 (1960), 77–84.Google Scholar
3.Boltjansky, V. G. and Gohberg, I.. Results and Problems in Combinatorial Geometry (Cambridge Univ. Press, Cambridge, 1985).Google Scholar
4.Boltjansky, V. G. and Soltan, P. S.. Combinatorial Geometry of Various Classes of Convex Sets (Shtiintsa, Kishinev, 1978).Google Scholar
5.Gohberg, I. and Markus, A. S.. A certain problem about the covering of convex sets with homothetic ones (in Russian). Izvestiya Mold Fil. Akad. Nauk SSSR, 76 (1960), 87–90.Google Scholar
6.Grőtschel, M., Lovász, L. and Schrijver, A.. Geometric Algorithms and Combinatorial Optimization (Springer-Verlag, Berlin-New York, 1989).Google Scholar
7.Hadwiger, H.. Ungelőste Probleme, Nr. 20. Elem. Math., 12 (1957), 121.Google Scholar
8.Hadwiger, H.. Ungelőste Probleme, Nr. 38. Elem. Math., 15 (1960), 130–131.Google Scholar
9.Lassak, M.. Solution of Hadwiger's covering problem for centrally symmetric convex bodies of E3. J. London Math. Soc. (2), 30 (1984), 501–511.Google Scholar
10.Lassak, M.. Covering plane convex bodies with smaller homothetical copies. Coll. Math. Soc. J. Bolyai, Vol. 48, Intuitive Geometry, (1985), 331–337.Google Scholar
11.Lassak, M.. Covering a plane convex body by four homothetical copies with the smallest positive ratio. Geom. Dedicate, 21 (1986), 151–167.Google Scholar
12.Lassak, M.. Covering the boundary of a convex set by tiles. Proc. Amer. Math. Soc, 104 (1988), 269–272.CrossRefGoogle Scholar
13.Levi, F. W.. Ein geometrisches Überdeckungsproblem. Arch. Math., 5 (1954), 476–478.CrossRefGoogle Scholar
14.Levi, F. W.. Überdeckung eines Eibereiches durch Parallelverschiebungen seines offenen Kerns. Arch. Math., 6 (1955), 369–370.CrossRefGoogle Scholar
15.McMullen, P. and Shephard, G. C.. Convexpolytopes and the upper bound conjecture (Cambridge Univ. Press, Cambridge, 1971).Google Scholar
16.Schramm, O.. Illuminating sets of contant width. Mathematika, 35 (1988), 180–189.CrossRefGoogle Scholar
17.Soltan, P. S.. Towards the problem of covering and illumination of convex sets (in Russian). Izvestiya Akad. Nauk. Mold. SSSR, (1963), 49–57.Google Scholar
18.Soltan, P. S. and Soltan, V. P.. On the X-raying of convex bodies (in Russian). Sov. Math. Dokl, 33 (1986), 42–44.Google Scholar