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Cross $ i$-sections of star bodies and dual mixed volumes


Authors: Songjun Lv and Gangsong Leng
Journal: Proc. Amer. Math. Soc. 135 (2007), 3367-3373
MSC (2000): Primary 52A30, 52A40
DOI: https://doi.org/10.1090/S0002-9939-07-08997-6
Published electronically: June 20, 2007
MathSciNet review: 2322769
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Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we establish an extension of Funk's section theorem. Our result has the following corollary: If $ K$ is a star body in $ \mathbb{R}^n$ whose central $ i$-slices have the same volume (with appropriate dimension) as the central $ i$-slices of a centered body $ M$, then the dual quermassintegrals satisfy $ \widetilde{W}_j(M)\leq \widetilde{W}_j(K)$, for any $ 0\leq j<n-i$, with equality if and only if $ K=M$. The case that $ K$ is a centered body implies Funk's section theorem.


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  • 1. A. D. Aleksandrov, On the theory of mixed volumes. II. New inequalities beteeen mixed volumes and their application, Mat. Sbornik N. S. 2 (1937), 1205-1238, Russian.
  • 2. G. D. Chakerian, E. Lutwak, Bodies with similar projections, Trans. Amer. Math. Soc. 349 (1997), 1811-1820. MR 1390034 (98a:52011)
  • 3. P. Funk, Über Flächen mit lauter geschlossenen geodätischen Linien, Math. Ann. 74 (1913), 278-300. MR 1511763
  • 4. R. J. Gardner, Geometric Tomography, New York, Cambridge Univ. Press, 1995. MR 1356221 (96j:52006)
  • 5. E. Grinberg, G. Zhang, Convolutions, transforms and convex bodies, Proc. London Math. Soc. 78 (1999), 77-115. MR 1658156 (99m:52009)
  • 6. P. R. Halmos, Measure Theory, Princeton, Van Nostrand, 1950. MR 0033869 (11:504d)
  • 7. S. Helgason, Groups and geometric analysis, Academic Press, 1984. MR 754767 (86c:22017)
  • 8. S. Helgason, The Radon transform, Birkhäuser, Boston, Second edition, 1999. MR 1723736 (2000m:44003)
  • 9. E. Lutwak, Intersection bodies and dual mixed volumes, Adv. Math. 71 (1988), 232-261. MR 963487 (90a:52023)
  • 10. E. Lutwak, Dual mixed volumes, Pacific J. Math. 58 (1975), 531-538. MR 0380631 (52:1528)
  • 11. E. Milman, Generalized intersection bodies, J. Funct. Anal. 240 (2006), 530-567. MR 2261694
  • 12. B. Rubin, Inversion formulas for the spherical Radon transform and the generalized cosine transform, Adv. Appl. Math. 29 (2002), 471-497. MR 1942635 (2004c:44006)
  • 13. R. Schneider and W. Weil, Zonoids and related topics, Convexity and its Applications (P.M. Gruber and J.M. Wills, Eds.), Birkhäuser, Basel, 1983, pp. 296-317. MR 731116 (85c:52010)
  • 14. W. Weil, Über den Vektorraum der Differenzen von Stützfunktionen konvexer Körper, Math. Nachr. 59 (1974), 353-369. MR 0341283 (49:6033)
  • 15. W. Weil, Kontinuierliche Linearkombination von Strecken, Math. Z. 148 (1976), 71-84. MR 0400052 (53:3887)
  • 16. W. Weil, Centrally symmetric convex bodies and distributions II, Israel J. Math. 32 (1979), 143-182. MR 531260 (80g:52003)
  • 17. G. Zhang, Centered bodies and dual mixed volumes, Trans. Amer. Math. Soc. 345 (1994), 777-801. MR 1254193 (95d:52008)
  • 18. G. Zhang, Sections of convex bodies, Amer. J. Math. 118 (1996), 319-340. MR 1385280 (97f:52015)

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Additional Information

Songjun Lv
Affiliation: Department of Mathematics, Shanghai University, Shanghai, People’s Republic of China 200444
Email: lvsongjun@126.com

Gangsong Leng
Affiliation: Department of Mathematics, Shanghai University, Shanghai, People’s Republic of China 200444
Email: gleng@staff.shu.edu.cn

DOI: https://doi.org/10.1090/S0002-9939-07-08997-6
Keywords: Star body, cross $i$-section, dual mixed volume, Radon transform
Received by editor(s): June 19, 2006
Published electronically: June 20, 2007
Additional Notes: This research was supported, in part, by NSFC Grant 10671117.
Communicated by: Jonathan M. Borwein
Article copyright: © Copyright 2007 American Mathematical Society

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