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Nakajima's Problem: Convex Bodies of Constant Width and Constant Brightness

Part of: General convexity

Published online by Cambridge University Press:  21 December 2009

Ralph Howard  and
Daniel Hug
Ralph Howard
Affiliation:
Department of Mathematics, University of South Carolina, Columbia, SC 29208, U.S.A. E-mail: [email protected]
Daniel Hug
Affiliation:
Fachbereich Mathematik, Universität Duisburg-Essen, Campus Essen, D-45117 Essen, Germany.
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Abstract

The kth projection function of a convex body K ⊂ ℝn assigns to any k-dimensional linear subspace of ℝn the k-volume of the orthogonal projection of K to that subspace. Let K and K0 be convex bodies in ℝn, and let K0 be centrally symmetric and satisfy a weak regularity and curvature condition (which includes all K0 with ∂K0 of class C2 with positive radii of curvature). Assume that K and K0 have proportional 1st projection functions (i.e., width functions) and proportional kth projection functions. For 2 ≤ k < (n + 1)/2 and for k = 3, n = 5, it is shown that K and K0 are homothetic. In the special case where K0 is a Euclidean ball, characterizations of Euclidean balls as convex bodies of constant width and constant k-brightness are thus obtained.

MSC classification

Secondary: 52A20: Convex sets in $n$ dimensions (including convex hypersurfaces)
Type
Research Article
Information
Mathematika , Volume 54 , Issue 1-2 , December 2007 , pp. 15 - 24
Copyright
Copyright © University College London 2007

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References

1Chakerian, G. D., Sets of constant relative width and constant relative brightness. Trans. Amer. Math. Soc. 129 (1967), 26–37.CrossRefGoogle Scholar
2Chakerian, G. D. and Groemer, H., Convex bodies of constant width. In Convexity and its Applications, Birkhäuser (Basel, 1983), 49–96.CrossRefGoogle Scholar
3Croft, H. T., Falconer, K. J. and Guy, R. K., Unsolved Problems in Geometry. Corrected reprint of the 1991 original Problem Books in Mathematics: Unsolved Problems in Intuitive Mathematics, II. Springer-Verlag (New York, 1994), xvi+198 pp.Google Scholar
4Gardner, R. J., Geometric Tomography (Encyclopedia of Mathematics and its Applications 58). Cambridge University Press (New York, 1995).Google Scholar
5Goodey, P. R., Schneider, R. and Weil, W., Projection functions of convex bodies. Intuitive Geometry (Budapest, 1995), Bolyai Soc. Math. Stud. 6. János Bolyai Math. Soc. (Budapest, 1997), 23–53.Google Scholar
6Heil, E. and Martini, H., Special convex bodies. In Handbook of Convex Geometry (ed. Gruber, P. M. and Wills, J. M.), North-Holland (Amsterdam, 1993), 347–385.CrossRefGoogle Scholar
7Howard, R., Convex bodies of constant width and constant brightness. Advances Math. 204 (2006), 241–261.CrossRefGoogle Scholar
8Howard, R. and Hug, D., Smooth convex bodies with proportional projection functions. Israel J. Math. (to appear).Google Scholar
9Hug, D., Curvature relations and affine surface area for a general convex body and its polar. Results Math. 29 (1996), 233–248.CrossRefGoogle Scholar
10Leichtweiß, K., Über einige Eigenschaften der Affinobefläche beliebiger konvexer Körper. Results Math. 13 (1988), 255–282.CrossRefGoogle Scholar
11Nakajima, S., Eine charakteristische Eigenschaft der Kugel. Jber. Deutsche Math.-Verein 35 (1926), 298–300.Google Scholar
12Schneider, R., Bestimmung konvexer Körper durch Krümmungsmaße. Comment. Math. Helvet. 54 (1979), 42–60.CrossRefGoogle Scholar
13Schneider, R., Convex Bodies: the Brunn-Minkowski Theory (Encyclopedia of Mathematics and its Applications 44), Cambridge University Press (Cambridge, 1993).CrossRefGoogle Scholar