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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Nakajima’s problem for general convex bodies
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by Daniel Hug PDF
Proc. Amer. Math. Soc. 137 (2009), 255-263 Request permission


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
  • MR Author ID: 363423
  • Email:
  • 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
  • © Copyright 2008 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 137 (2009), 255-263
  • MSC (2000): Primary 52A20; Secondary 52A39, 53A05
  • DOI:
  • MathSciNet review: 2439448