The Shephard type problems and monotonicity for L p -mixed centroid body

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Abstract

Lutwak and Zhang proposed the notion of L p -centroid body. Further, Ma gave the definition of L p -mixed centroid body, and obtained affirmative form for the Shephard type problems of L p -mixed centroid body. In this article, we first give another affirmative form of the Shephard type problems for L p -mixed centroid body, meanwhile, obtain its negative form. Next, we also give an extension of the generalized Funk’s section theorem for L p -mixed centroid body. Finally, we establish two monotonicity inequalities of L p -mixed centroid body.

Key words

Lp-mixed centroid body Shephard type problems monotonicity 
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References

  1. 1.
    R. J. Gardner, Geometric Tomography, Second ed., Cambridge Univ. Press, Cambridge, 2006.CrossRefMATHGoogle Scholar
  2. 2.
    E. Grinberg and G. Y. Zhang, Convolutions transforms and convex bodies, Pron. London Math. Soc, 78(3) (1999), 77–115.CrossRefMathSciNetGoogle Scholar
  3. 3.
    C. Haberl, L p-intersection bodies, Adv. Math., 217 (2008), 2599–2624.CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    G. S. Leng, S. Lv and J. Yuan, Dual L p-centroid bodies and the Fourier transform, Acta Math. Sinica-Chinese Edition, 50 (2007), 1419–1424.MATHGoogle Scholar
  5. 5.
    E. Lutwak and G. Y. Zhang, Blaschke-Santaló inequalities, J. Differential Geom., 47(1) (1997), 1–16.MATHMathSciNetGoogle Scholar
  6. 6.
    E. Lutwak, D. Yang and G. Y. Zhang, L p-affine isoperimetric inequalities, 7. Differential Geom., 56(1) (2000), 111–132.MATHMathSciNetGoogle Scholar
  7. 7.
    E. Lutwak, D. Yang and G. Y. Zhang, The Cramer-Rao inequality for star bodies, Duke. Math. J., 112(1) (2002), 59–81.CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    E. Lutwak, The Brunn-Minkowski-Firey theory I: mixed volumes and the Minkowski problem, J. Differential Geom., 38(1) (1993), 131–150.MATHMathSciNetGoogle Scholar
  9. 9.
    E. Lutwak, The Brunn-Minkowski-Firey theory II: affine and geominimal surface areas, Adv. Math., 118(2) (1996), 244–294.CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    E. Lutwak, Dual mixed volumes, Pacific J. Math., 58 (1975), 531–538.CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    S. Lv and G. S. Leng, The L p-curvature images of convex bodies and L p-projection bodies, Proc. Indian Acad. Sci. Math. Sci., 18 (2008), 413–424.CrossRefMathSciNetGoogle Scholar
  12. 12.
    S. Lv and G. S. Leng, Cross i-sections of star bodies and dual mixed volumes, Proc. Amer. Math. Soc., 135 (2007), 3367–3373.CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    T. Y. Ma, On L p-mixed centroid bodies and dual L p-mixed centroid bodies, Acta Math. Sinica, 53(2) (2010), 301–314.MATHMathSciNetGoogle Scholar
  14. 14.
    D. Ryabogin and A. Zvavitch, The Fourier transform and Firey projections of convex bodies, Indiana Univ. Math. J., 53 (2004), 667–682.CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    R. Schneider, Convex Bodies: The Brunn-Minkowski Theory, Cambridge Univ., Press, Cambridge, 1993.CrossRefMATHGoogle Scholar
  16. 16.
    W. D. Wang, F. H. Lu and G. S. Leng, A type of monotonicity on the L p-centroid body and L p-projection body, Math. Inequal. & Appl., 8(4) (2005), 735–742.MATHMathSciNetGoogle Scholar
  17. 17.
    W. D. Wang and G. S. Leng, On the monotonicity of L p-centroid body, J. Sys. Sci. & Math. Scis., 28(2) (2008), 154–162, (in Chinese).MATHMathSciNetGoogle Scholar
  18. 18.
    W. D. Wang and G. S. Leng, L p-dual mixed quermassintegrals, Indian J. Pure Appl. Math., 36(4) (2005), 177–188.MATHMathSciNetGoogle Scholar
  19. 19.
    W. D. Wang and G. S. Leng, The Petty projection inequality for L p-mixed projection bodies, Acta Mathematica Sinica, 23(8) (2007), 1485–1494.CrossRefMATHMathSciNetGoogle Scholar
  20. 20.
    J. Yuan, L. Z. Zhao and G. S. Leng, Inequalities for L p-centroid body, Tai wan. J. Math., 29(2) (2008), 209–220.Google Scholar
  21. 21.
    J. Yuan and W. Chueng, L p-intersection bodies, J. Math. Anal. Appl., 338 (2008), 1431–1439.CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© The Indian National Science Academy 2014

Authors and Affiliations

  1. 1.Department of MathematicsChina Three Gorges UniversityYichangChina

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