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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: howard@math.sc.edu
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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