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On Star Duality of Mixed Intersection Bodies

Abstract

A new kind of duality between intersection bodies and projection bodies is presented. Furthermore, some inequalities for mixed intersection bodies are established. A geometric inequality is derived between the volumes of star duality of star bodies and their associated mixed intersection integral.

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References

  1. 1.

    Lutwak E: Intersection bodies and dual mixed volumes. Advances in Mathematics 1988,71(2):232–261. 10.1016/0001-8708(88)90077-1

    MathSciNet  Article  MATH  Google Scholar 

  2. 2.

    Zhang GY: Centered bodies and dual mixed volumes. Transactions of the American Mathematical Society 1994,345(2):777–801. 10.2307/2154998

    MathSciNet  Article  MATH  Google Scholar 

  3. 3.

    Santaló L: Un invariante afin para los cuerpos convexos del espacio dedimeniones. Portugaliae Mathematica 1949,8(4):155–161.

    Google Scholar 

  4. 4.

    Moszyńska M: Quotient star bodies, intersection bodies, and star duality. Journal of Mathematical Analysis and Applications 1999,232(1):45–60. 10.1006/jmaa.1998.6238

    MathSciNet  Article  MATH  Google Scholar 

  5. 5.

    Haberl C, Ludwig M: A characterization ofintersection bodies. International Mathematics Research Notices 2006, 2006: 29 pages.

    MathSciNet  MATH  Google Scholar 

  6. 6.

    Kalton NJ, Koldobsky A: Intersection bodies and-spaces. Advances in Mathematics 2005,196(2):257–275. 10.1016/j.aim.2004.09.002

    MathSciNet  Article  MATH  Google Scholar 

  7. 7.

    Klain DA: Star valuations and dual mixed volumes. Advances in Mathematics 1996,121(1):80–101. 10.1006/aima.1996.0048

    MathSciNet  Article  MATH  Google Scholar 

  8. 8.

    Klain DA: Invariant valuations on star-shaped sets. Advances in Mathematics 1997,125(1):95–113. 10.1006/aima.1997.1601

    MathSciNet  Article  MATH  Google Scholar 

  9. 9.

    Koldobsky A: A functional analytic approach to intersection bodies. Geometric and Functional Analysis 2000,10(6):1507–1526. 10.1007/PL00001659

    MathSciNet  Article  MATH  Google Scholar 

  10. 10.

    Ludwig M: Minkowski valuations. Transactions of the American Mathematical Society 2005,357(10):4191–4213. 10.1090/S0002-9947-04-03666-9

    MathSciNet  Article  MATH  Google Scholar 

  11. 11.

    Ludwig M: Intersection bodies and valuations. American Journal of Mathematics 2006.,128(6):

  12. 12.

    Lutwak E: Mixed projection inequalities. Transactions of the American Mathematical Society 1985,287(1):91–106. 10.1090/S0002-9947-1985-0766208-7

    MathSciNet  Article  MATH  Google Scholar 

  13. 13.

    Gardner RJ: Geometric Tomography, Encyclopedia of Mathematics and Its Applications. Volume 58. Cambridge University Press, Cambridge, UK; 1995.

    Google Scholar 

  14. 14.

    Schneider R: Convex Bodies: The Brunn-Minkowski Theory, Encyclopedia of Mathematics and Its Applications. Volume 44. Cambridge University Press, Cambridge, UK; 1993.

    Google Scholar 

  15. 15.

    Lutwak E: Dual mixed volumes. Pacific Journal of Mathematics 1975,58(2):531–538.

    MathSciNet  Article  MATH  Google Scholar 

  16. 16.

    Lutwak E: Centroid bodies and dual mixed volumes. Proceedings of the London Mathematical Society. Third Series 1990,60(2):365–391. 10.1112/plms/s3-60.2.365

    MathSciNet  Article  MATH  Google Scholar 

  17. 17.

    Moszyńska M: Selected Topics in Convex Geometry. Springer, New York, NY, USA; 2005.

    Google Scholar 

  18. 18.

    Hardy H, Littlewood JE, Pólya G: Inequalities. Cambridge University Press, London, UK; 1934.

    Google Scholar 

  19. 19.

    Busemann H: Volume in terms of concurrent cross-sections. Pacific Journal of Mathematics 1953, 3: 1–12.

    MathSciNet  Article  MATH  Google Scholar 

  20. 20.

    Petty CM: Isoperimetric problems. In Proceedings of the Conference on Convexity and Combinatorial Geometry, 1971, Norman, Okla, USA. Department of Mathematics, University of Oklahoma; 26–41.

    Google Scholar 

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Correspondence to Lu Fenghong.

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Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Fenghong, L., Weihong, M. & Gangsong, L. On Star Duality of Mixed Intersection Bodies. J Inequal Appl 2007, 039345 (2007). https://doi.org/10.1155/2007/39345

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Keywords

  • Mixed Intersection
  • Intersection Body
  • Projection Body
  • Star Body
  • Geometric Inequality
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