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Some inequalities for the L p -mixed quermassintegrals

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Wuhan University Journal of Natural Sciences

Abstract

In this paper, a cyclic inequality and a monotonic inequality of L p -mixed quermassintegrals are established. Meanwhile, we obtain an inequality for L p -mixed quermassintegrals of convex bodies and its polar.

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References

  1. Firey W J. p-Means of convex bodies [J]. Math Scand, 1962, 10(1): 17–24.

    MathSciNet  MATH  Google Scholar 

  2. Lutwak E. The Brunn-Minkowski-Firey theory I: Mixed volumes and the Minkowski problem [J]. J Differential Geom, 1993, 38(1): 131–150.

    MathSciNet  MATH  Google Scholar 

  3. Lutwak E. The Brunn-Minkowski-Firey theory II:Affine and geominimal surface areas [J]. Adv Math, 1996, 118(2): 244–294.

    Article  MathSciNet  MATH  Google Scholar 

  4. Lutwak E, Yang D, Zhang G Y. L p affine isoperimetric inequalities [J]. J Differential Geom, 2000, 56(1): 111–132.

    MathSciNet  MATH  Google Scholar 

  5. Campi S, Gronchi P. The L p-Busemann-Petty centroid inequality [J]. Adv Math, 2002, 167(1): 128–141.

    Article  MathSciNet  MATH  Google Scholar 

  6. Haberl C, Ludwig M. A characterization of L p intersection bodies [C]// International Mathematics Research Notices, Art ID 10548, New York: Springer-Vertag, 2006.

    Google Scholar 

  7. Haberl C. L p intersection bodies [J]. Adv Math, 2008, 217(6): 2599–2624.

    Article  MathSciNet  MATH  Google Scholar 

  8. Haberl C, Schuster F. General L p affine isoperimetric inequalities [J]. J Differential Geom, 2009, 83(1): 1–26.

    MathSciNet  MATH  Google Scholar 

  9. Grinberg E, Zhang G Y. Convolutions, transforms, and convex bodies [J]. Proc London Math Soc, 1999, 78(1): 77–115.

    Article  MathSciNet  Google Scholar 

  10. Lutwak E, Zhang G Y. Blaschke-Santal Ȯ inequalities [J]. J Differential Geom, 1997, 47(1): 1–16.

    MathSciNet  MATH  Google Scholar 

  11. Lv S J, Leng G S. The L p-curvature images of convex bodies and L p-projection bodies [J]. Proc Indian Acad Sci (Math Sci), 2008, 118(3): 413–424.

    Article  MathSciNet  MATH  Google Scholar 

  12. Wang W D, Leng G S. The Petty projection inequality for L p-mixed projection bodies [J]. Acta Math Sinica (English Series), 2007, 23(8): 1485–1494.

    Article  MathSciNet  MATH  Google Scholar 

  13. Wang W D, Leng G S. On the L p-versions of the Petty’s conjectured projection inequality and applications [J]. Taiwan J Math, 2008, 12(5): 1067–1086.

    MathSciNet  MATH  Google Scholar 

  14. Wang Weidong. On reverses of the L p-Busemann-Petty centroid inequality and its applications [J]. Wuhan University Journal of Natural Sciences, 2010, 15(4): 292–296.

    Article  MathSciNet  MATH  Google Scholar 

  15. Wang W D, Leng G S. L p-mixed affine surface area [J]. J Math Anal Appl, 2007, 335(1): 341–354.

    Article  MathSciNet  MATH  Google Scholar 

  16. Wang Weidong, Wei D J, Xiang Y. Some inequalities for the L p-curvature image [J]. J Inequal Appl, 2009, 6: 1–12.

    MathSciNet  Google Scholar 

  17. Wang W D, Qi C. L p-dual geominimal surface area [J]. J Inequal Appl, 2011, 6: 1–10.

    Google Scholar 

  18. Wang W, He B W. L p-dual affine surface area [J]. J Math Anal Appl, 2008, 348(2): 746–751.

    Article  MathSciNet  MATH  Google Scholar 

  19. Werner E, Ye D. Inequalities for mixed p-affine surface area [J]. Math Ann, 2010, 347(3): 703–737.

    Article  MathSciNet  MATH  Google Scholar 

  20. Gardner R J, Zhang G Y. Affine inequalities and radial mean bodies [J]. Amer J Math, 1998, 120(2): 505–528.

    Article  MathSciNet  MATH  Google Scholar 

  21. Gardner R J, Giannopoulos A. p-Cross-section bodies [J]. Indiana U Math J, 1999, 48(2): 593–613.

    Article  MathSciNet  MATH  Google Scholar 

  22. Lutwak E, Yang D, Zhang G Y. Sharp affine L p Sobolev inequalities [J]. J Differential Geom, 2002, 52(1): 17–38.

    MathSciNet  Google Scholar 

  23. Lutwak E, Yang D, Zhang G Y. L p John ellipsoids [J]. Proc London Math Soc, 2005, 90(2): 497–520.

    Article  MathSciNet  MATH  Google Scholar 

  24. Wang W D, Leng G S. L p-dual mixed [J]. Indian J Pure Appl Math, 2005, 36(4): 177–188.

    MathSciNet  MATH  Google Scholar 

  25. Ghandehari M. Polar duals of convex bodies [J]. Proc Amer Math Soc, 1991, 113(3): 709–808.

    Article  MathSciNet  Google Scholar 

  26. Gardner R J. Geometric Tomography [M]. Second Edition Cambridge: Cambridge Univ Press, 2006.

    Google Scholar 

  27. Hardy G H, Littlewood J E, Pólya G. Inequalities [M]. Cambridge: Cambrige University Press, 1934.

    Google Scholar 

  28. Zhao C J, Chen L Y, Cheung W S. Polar duals of convex and star bodies [J]. J Inequal Appl, 2012, 2: 1–10.

    MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, China Three Gorges University, Yichang, 443002, Hubei, China

    Bo Wei & Weidong Wang

Authors
  1. Bo Wei
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  2. Weidong Wang
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Corresponding author

Correspondence to Weidong Wang.

Additional information

Foundation item: Supported by the National Natural Science Foundation of China (10671117) and Science Foundation of China Three Gorges University

Biography: WEI Bo, male, Master candidate, research direction: convex geometric analysis.

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Wei, B., Wang, W. Some inequalities for the L p -mixed quermassintegrals. Wuhan Univ. J. Nat. Sci. 18, 233–236 (2013). https://doi.org/10.1007/s11859-013-0920-1

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  • DOI: https://doi.org/10.1007/s11859-013-0920-1

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