## Nakajima’s problem for general convex bodies

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- Proc. Amer. Math. Soc.
**137**(2009), 255-263 Request permission

## 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

- A. D. Alexandroff,
*Almost everywhere existence of the second differential of a convex function and some properties of convex surfaces connected with it*, Leningrad State Univ. Annals [Uchenye Zapiski] Math. Ser.**6**(1939), 3–35 (Russian). MR**0003051** - Christina Bauer,
*Intermediate surface area measures and projection functions of convex bodies*, Arch. Math. (Basel)**64**(1995), no. 1, 69–74. MR**1305662**, DOI 10.1007/BF01193552 - Stefano Campi,
*Reconstructing a convex surface from certain measurements of its projections*, Boll. Un. Mat. Ital. B (6)**5**(1986), no. 3, 945–959 (English, with Italian summary). MR**871707** - G. D. Chakerian,
*Sets of constant relative width and constant relative brightness*, Trans. Amer. Math. Soc.**129**(1967), 26–37. MR**212678**, DOI 10.1090/S0002-9947-1967-0212678-1 - G. D. Chakerian and H. Groemer,
*Convex bodies of constant width*, Convexity and its applications, Birkhäuser, Basel, 1983, pp. 49–96. MR**731106** - G. D. Chakerian and E. Lutwak,
*Bodies with similar projections*, Trans. Amer. Math. Soc.**349**(1997), no. 5, 1811–1820. MR**1390034**, DOI 10.1090/S0002-9947-97-01760-1 - Hallard T. Croft, Kenneth J. Falconer, and Richard K. Guy,
*Unsolved problems in geometry*, Problem Books in Mathematics, Springer-Verlag, New York, 1991. Unsolved Problems in Intuitive Mathematics, II. MR**1107516**, DOI 10.1007/978-1-4612-0963-8 - William J. Firey,
*Convex bodies of constant outer $p$-measure*, Mathematika**17**(1970), 21–27. MR**267465**, DOI 10.1112/S0025579300002667 - Richard J. Gardner,
*Geometric tomography*, Encyclopedia of Mathematics and its Applications, vol. 58, Cambridge University Press, Cambridge, 1995. MR**1356221** - R. J. Gardner and A. Volčič,
*Tomography of convex and star bodies*, Adv. Math.**108**(1994), no. 2, 367–399. MR**1296519**, DOI 10.1006/aima.1994.1075 - P. Goodey, R. Schneider, and W. Weil,
*Projection functions of convex bodies*, Intuitive geometry (Budapest, 1995) Bolyai Soc. Math. Stud., vol. 6, János Bolyai Math. Soc., Budapest, 1997, pp. 23–53. MR**1470754** - Paul Goodey, Rolf Schneider, and Wolfgang Weil,
*On the determination of convex bodies by projection functions*, Bull. London Math. Soc.**29**(1997), no. 1, 82–88. MR**1416411**, DOI 10.1112/S0024609396001968 - P. Goodey, R. Howard,
*Examples and structure of smooth convex bodies of constant $k$-brightness*(in preparation). - Paul Goodey and Gaoyong Zhang,
*Inequalities between projection functions of convex bodies*, Amer. J. Math.**120**(1998), no. 2, 345–367. MR**1613642** - Eric Grinberg and Gaoyong Zhang,
*Convolutions, transforms, and convex bodies*, Proc. London Math. Soc. (3)**78**(1999), no. 1, 77–115. MR**1658156**, DOI 10.1112/S0024611599001653 - E. Heil and H. Martini,
*Special convex bodies*, Handbook of convex geometry, Vol. A, B, North-Holland, Amsterdam, 1993, pp. 347–385. MR**1242985** - Ralph Howard,
*Convex bodies of constant width and constant brightness*, Adv. Math.**204**(2006), no. 1, 241–261. MR**2233133**, DOI 10.1016/j.aim.2005.05.015 - Ralph Howard and Daniel Hug,
*Smooth convex bodies with proportional projection functions*, Israel J. Math.**159**(2007), 317–341. MR**2342484**, DOI 10.1007/s11856-007-0049-z - R. Howard, D. Hug,
*Nakajima’s problem: convex bodies of constant width and constant brightness*, Mathematika (to appear). - Daniel Hug,
*Contributions to affine surface area*, Manuscripta Math.**91**(1996), no. 3, 283–301. MR**1416712**, DOI 10.1007/BF02567955 - Daniel Hug,
*Absolute continuity for curvature measures of convex sets. II*, Math. Z.**232**(1999), no. 3, 437–485. MR**1719698**, DOI 10.1007/PL00004765 - P. McMullen,
*On the inner parallel body of a convex body*, Israel J. Math.**19**(1974), 217–219. MR**367810**, DOI 10.1007/BF02757715 - S. Nakajima,
*Eine charakteristische Eigenschaft der Kugel*, Jber. Deutsche Math.-Verein**35**(1926), 298–300. - Rolf Schneider,
*Convex bodies: the Brunn-Minkowski theory*, Encyclopedia of Mathematics and its Applications, vol. 44, Cambridge University Press, Cambridge, 1993. MR**1216521**, DOI 10.1017/CBO9780511526282 - Rolf Schneider,
*Polytopes and Brunn-Minkowski theory*, Polytopes: abstract, convex and computational (Scarborough, ON, 1993) NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., vol. 440, Kluwer Acad. Publ., Dordrecht, 1994, pp. 273–299. MR**1322067**

## 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: 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