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Nakajima's problem for general convex bodies

Author: Daniel Hug
Journal: Proc. Amer. Math. Soc. 137 (2009), 255-263
MSC (2000): Primary 52A20; Secondary 52A39, 53A05
Published electronically: July 8, 2008
MathSciNet review: 2439448
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Abstract: For a convex body $ K\subset\mathbb{R}^n$, the $ k$th projection function of $ K$ assigns to any $ k$-dimensional linear subspace of $ \mathbb{R}^n$ the $ k$-volume of the orthogonal projection of $ K$ to that subspace. Let $ K$ and $ K_0$ be convex bodies in $ \mathbb{R}^n$, and let $ K_0$ be centrally symmetric and satisfy a weak regularity assumption. Let $ i,j\in\mathbb{N}$ be such that $ 1\le i<j\le n-2$ with $ (i,j)\neq (1,n-2)$. Assume that $ K$ and $ K_0$ have proportional $ i$th projection functions and proportional $ j$th projection functions. Then we show that $ K$ and $ K_0$ are homothetic. In the particular case where $ K_0$ is a Euclidean ball, we thus obtain characterizations of Euclidean balls as convex bodies having constant $ i$-brightness and constant $ j$-brightness. This special case solves Nakajima's problem in arbitrary dimensions and for general convex bodies for most indices $ (i,j)$.

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Additional Information

Daniel Hug
Affiliation: Fakultät für Mathematik, Institut für Algebra und Geometrie, Universität Karlsruhe (TH), KIT, D-76128 Karlsruhe, Germany

Received by editor(s): July 12, 2007
Received by editor(s) in revised form: November 20, 2007
Published electronically: July 8, 2008
Additional Notes: The author was supported in part by the European Network PHD, FP6 Marie Curie Actions, RTN, Contract MCRN-511953.
Communicated by: N. Tomczak-Jaegermann
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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