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Illuminating sets of constant width

Part of: General convexity

Published online by Cambridge University Press:  26 February 2010

Oded Schramm
Oded Schramm
Affiliation:
Mathematics Department, Fine Hall, Princeton University, Princeton NJ 08544, USA.
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Abstract

The problem of illuminating the boundary of sets having constant width is considered and a bound for the number of directions needed is given. As a corollary, an estimate for Borsuk's partition problem is inferred. Also, the illumination number of sufficiently symmetric strictly convex bodies is determined.

MSC classification

Secondary: 52A20: Convex sets in $n$ dimensions (including convex hypersurfaces)
Type
Research Article
Information
Mathematika , Volume 35 , Issue 2 , December 1988 , pp. 180 - 189
Copyright
Copyright © University College London 1988

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References

1.Artin, E.. The Gamma Function, translated from German (Holt, Rinehart and Winston, 1964).Google Scholar
2.Boltjansky, V. G.. The problem of the illumination of the boundary of a convex body. In Russian. Izvestiya Moldavskogo filiala Akademii Nauk SSSR, 10 (1960), 77–84.Google Scholar
3.Boltjansky, V. G. and Gohberg, I. Ts.. Results and problems in combinatorial geometry, translated from Russian (Cambridge Univ. Press, Cambridge, 1985).CrossRefGoogle Scholar
4.Chakerian, G. D. and Groemer, H.. Convex bodies of constant width, in “Convexity and its applications,” ed. by Gruber, P. M. and Wills, J. M. (Birkhäuser, Basel, 1983).Google Scholar
5.Eggleston, H. G.. Convexity (Cambridge Univ. Press, Cambridge, 1958).CrossRefGoogle Scholar
6.Grove, L. C. and Benson, C. T.. Finite Reflection Groups (Springer, New York, 1985).CrossRefGoogle Scholar
7.Grünbaum, B.. Borsuk's problem and related questions. Proc. Symp. Pure Math., 7 (1963), 271–284.Google Scholar
8.Grünbaum, B.. Convex Polytopes (Wiley, New York, 1967).Google Scholar
9.Rogers, C. A.. Covering a sphere with spheres. Mathematika, 10 (1963), 157–164.Google Scholar
10.Rogers, C. A.. Symmetrical sets of constant width and their partitions. Mathematika, 18 (1971), 105–111.CrossRefGoogle Scholar
11.Schramm, O.. On the volume of sets having constant width. To appear, Israel J. Math.Google Scholar