## Nakajima’s problem for general convex bodies

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- by Daniel Hug
- Proc. Amer. Math. Soc.
**137**(2009), 255-263 - DOI: https://doi.org/10.1090/S0002-9939-08-09432-X
- Published electronically: July 8, 2008
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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 $(i,j)$.## References

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## Bibliographic 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: daniel.hug@kit.edu
- 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: https://doi.org/10.1090/S0002-9939-08-09432-X
- MathSciNet review: 2439448