Acta Mathematica Hungarica

, Volume 159, Issue 1, pp 211–228 | Cite as

On the reverse Orlicz–Lorentz Busemann–Petty centroid inequality

  • Y. FengEmail author
  • T. Ma


This paper studies the extrema of some affine invariant functionals related to the volume of the Orlicz–Lorentz centroid body introduced by Nguyen. We obtain some variants of the Orlicz–Lorentz Busemann–Petty centroid inequality, and also prove the reverse form of these inequalities in the two-dimensional case.

Key words and phrases

Orlicz–Lorentz centroid body Orlicz–Lorentz Busemann–Petty centroid inequality the shadow system 

Mathematics Subject Classification

52A20 52A40 


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  1. 1.
    Bisztriczky, T., Böröczky Jr., K.: About the centroid body and the ellipsoid of inertia. Mathematika 48, 1–13 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Campi, S., Gronchi, P.: The \(L_p\) Busemann-Petty centroid inequality. Adv. Math. 167, 128–142 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Campi, S., Gronchi, P.: On the reverse \(L_p\) Busemann-Petty centroid inequality. Mathematika 49, 1–11 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Chen, W.: \(L_p\) Minkowski problem with not necessarily positive data. Adv. Math. 201, 77–89 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Chen, F., Zhou, J., Yang, C.: On the reverse Orlicz Busemann-Petty centroid inequality. Adv. in Appl. Math. 47, 820–828 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Chou, K., Wang, X.: The \(L_p\) Minkowski problem and the Minkowski problem in centroaffine geometry. Adv. Math. 205, 33–83 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Dafnis, N., Paouris, G.: Small ball probability estimates, \(\psi _2\)-behavior and the hyperplane conjecture. J. Funct. Anal. 258, 1933–1964 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Fleury, B., Guédon, O., Paouris, G.: A stability result for mean width of \(L_p\) centroid bodies. Adv. Math. 214, 865–877 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Gardner, R.J.: Geometric Tomography, 2nd edn. Cambridge Univ, Press (New York (2006)CrossRefzbMATHGoogle Scholar
  10. 10.
    Gardner, R.J., Hug, D., Weil, W.: The Orlicz-Brunn-Minkowski theory: a general framework, additions, and inequalities. J. Differential Geom. 97, 427–476 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Gardner, R.J., Hug, D., Weil, W., Ye, D.: The dual Orlicz-Brunn-Minkowski theory. J. Math. Anal. Appl. 430, 810–829 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Giannopoulos, A., Stavrakakis, P., Tsolomitis, A., Vritsiou, B.H.: Geometry of the \(L_p\) centroid bodies of an isotropic log-concave measure. Trans. Amer. Math. Soc. 367, 4569–4593 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Guédon, O., Milman, E.: Interpolating thin-shell and sharp large-deviation estimates for isotropic log-concave measures. Geom. Funct. Anal. 21, 1043–1068 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Haberl, C., Schuster, F.E., Xiao, J.: An asymmetric affine Pólya-Szegő principle. Math. Ann. 352, 517–542 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Hu, C., Ma, X., Shen, C.: On the Christoffel-Minkowski problem of Fiery's \(p\)-sum. Calc. Var. Partial Differential Equations 21, 137–155 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Huang, Q., He, B.: On the Orlicz Minkowski problem for polytopes. Discrete Comput. Geom. 48, 281–297 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Klartag, B., Milman, E.: Centroid bodies and the logarithmic Laplace transform - A unified approach. J. Funct. Anal. 262, 10–34 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Li, A.J., Leng, G.: A new proof of the Orlicz Busemann-Petty centroid inequality. Proc. Amer. Math. Soc. 139, 1473–1481 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Lorentz, G.G.: Some new function spaces. Ann. Math. 51, 37–55 (1950)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Lorentz, G.G.: On the theory of spaces \(\Lambda \). Pacific J. Math. 1, 411–429 (1951)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Ludwig, M.: Ellipsoids and matrix-valued valuations. Duke Math. J. 119, 159–188 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Ludwig, M., Reitzner, M.: A classification of SL(n) invariant valuations. Ann. of Math. 172, 1223–1271 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Lutwak, E.: On some affine isoperimetric inequalities. J. Differential Geom. 23, 1–13 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Lutwak, E.: Centroid bodies and dual mixed volumes. Proc. London Math. Soc. 60, 365–391 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    E. Lutwak, The Brunn–Minkowski–Firey theory. I. Mixed volumes and the Minkowski problem, J. Differential Geom., 38 (1993), 131–150Google Scholar
  26. 26.
    E. Lutwak, The Brunn–Minkowski–Firey theory. II. Affine and geominimal surface areas, Adv. Math., 118 (1996), 244-294Google Scholar
  27. 27.
    Lutwak, E., Yang, D., Zhang, G.: A new ellipsoid associated with convex bodies. Duke Math. J. 104, 375–390 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Lutwak, E., Yang, D., Zhang, G.: On the \(L_p\) Minkowski Problem. Trans. Amer. Math. Soc. 356, 4359–4370 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Lutwak, E., Yang, D., Zhang, G.: \(L_p\) affine isoperimetric inequalities. J. Differential Geom. 56, 111–132 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Lutwak, E., Yang, D., Zhang, G.: Orlicz centroid bodies. J. Differential Geom. 84, 365–387 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Lutwak, E., Yang, D., Zhang, G.: Orlicz projection bodies. Adv. Math. 223, 220–242 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Nguyen, V.H.: Orlicz-Lorentz centroid bodies. Adv. in Appl. Math. 92, 99–121 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Paouris, G.: On the \(\psi _2\)-behavior of linear functionals on isotropic convex bodies. Stud. Math. 168, 285–299 (2005)CrossRefzbMATHGoogle Scholar
  34. 34.
    Paouris, G.: Concentration of mass on convex bodies. Geom. Funct. Anal. 16, 1021–1049 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Paouris, G.: Concentration of mass on isotropic convex bodies. C. R. Math. Acad. Sci. Paris 342, 179–182 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Paouris, G.: Small ball probability estimates for log-concave measures. Trans. Amer. Math. Soc. 364, 287–308 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Petty, C.M.: Centroid surfaces. Pacific J. Math. 11, 1535–1547 (1961)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    C. M. Petty, Ellipsoids, in: Convexity and its Applications (P. M. Gruber and J. M. Wills, eds.), Birkhäuser (Basel, 1983), pp. 264–276Google Scholar
  39. 39.
    Rogers, C.A., Shephard, G.C.: Some extremal problems for convex bodies. Mathematika 5, 93–102 (1958)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Schneider, R.: Convex Bodies: The Brunn-Minkowski Theory, 2nd edn. Cambridge Univ, Press (New York (2014)zbMATHGoogle Scholar
  41. 41.
    Shephard, G.C.: Shadow systems of convex bodies. Israel J. Math. 2, 229–236 (1964)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Stancu, A.: The discrete planar \(L_0\) Minkowski problem. Adv. Math. 167, 160–174 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Stancu, A.: On the number of solutions to the discrete two-dimensional \(L_0\) Minkowski problem. Adv. Math. 180, 290–323 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Trudinger, N.S., Wang, X.: The affine plateau problem. J. Amer. Math. Soc. 18, 253–289 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Xi, D., Jin, H., Leng, G.: The Orlicz-Brunn-Minkowski inequality. Adv. Math. 260, 350–374 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Ye, D.: Dual Orlicz-Brunn-Minkowski theory: Dual Orlicz \(L_\phi \) affine and geominimal surface areas. J. Math. Anal. Appl. 443, 352–371 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Zhu, G.: The Orlicz centroid inequality for star bodies. Adv. in Appl. Math. 48, 432–445 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Zhu, B., Zhou, J., Xu, W.: Dual Orlicz-Brunn-Minkowski theory. Adv. Math. 264, 700–725 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Zou, D., Xiong, G.: Orlicz-John ellipsoids. Adv. Math. 265, 132–168 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    Zou, D., Xiong, G.: Orlicz-Legendre ellipsoids. J. Geom. Anal. 26, 2474–2502 (2016)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2019

Authors and Affiliations

  1. 1.Department of MathematicsShanghai UniversityShanghaiChina
  2. 2.School of Mathematics and StatisticsHexi UniversityZhangyeChina

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