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The Log-Aleksandrov–Fenchel Inequality

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Abstract

In this paper, we first propose and establish a new Aleksandrov–Fenchel type inequality involving logarithms by introducing two new concepts of mixed volume measure and \(L_{p}\)-multiple mixed volume measure, and using the \(L_{p}\)-Aleksandrov–Fenchel inequality for the \(L_{p}\)-multiple mixed volumes. The new Aleksandrov–Fenchel inequality involving logarithms in special case yields the classical Aleksandrov–Fenchel inequality, and four recent logarithmic Minkowski inequalities, which are logarithmic Minkowski inequality for mixed volumes, \(L_{p}\)-mixed volumes, quermassintegrals and p-mixed quermassintegrals, respectively. Moreover, the logarithmic Aleksandrov–Fenchel inequality is also derived.

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References

  1. Böröczky, K.J., Lutwak, E., Yang, D., Zhang, G.: The log-Brunn–Minkowski inequality. Adv. Math. 231, 1974–1997 (2012)

    MathSciNet  MATH  Google Scholar 

  2. Stancu, A.: The logarithmic Minkowski inequality for non-symmetric convex bodies. Adv. Appl. Math. 73, 43–58 (2016)

    MathSciNet  MATH  Google Scholar 

  3. Zhu, G.: The logarithmic Minkowski problem for polytopes. Adv. Math. 262, 909–931 (2014)

    MathSciNet  MATH  Google Scholar 

  4. Fathi, M., Nelson, B.: Free Stein kernels and an improvement of the free logarithmic Sobolev inequality. Adv. Math. 317, 193–223 (2017)

    MathSciNet  MATH  Google Scholar 

  5. Colesanti, A., Cuoghi, P.: The Brunn–Minkowski inequality for the \(n\)-dimensional logarithmic capacity of convex bodies. Potential Math. 22, 289–304 (2005)

    MathSciNet  MATH  Google Scholar 

  6. Hou, S., Xiao, J.: A mixed volumetry for the anisotropic logarithmic potential. J. Geom. Anal. 28, 2018–2049 (2018)

    MathSciNet  MATH  Google Scholar 

  7. Henk, M., Pollehn, H.: On the log-Minkowski inequality for simplices and parallelepipeds. Acta Math. Hung. 155, 141–157 (2018)

    MathSciNet  MATH  Google Scholar 

  8. Wang, W., Liu, L.: The dual log-Brunn–Minkowski inequality. Taiwan. J. Math. 20, 909–919 (2016)

    MATH  Google Scholar 

  9. Wang, W., Feng, M.: The log-Minkowski inequalities for quermassintegrals. J. Math. Inequal. 11, 983–995 (2017)

    MathSciNet  MATH  Google Scholar 

  10. Li, C., Wang, W.: Log-Minkowski inequalities for the \(L_{p}\)-mixed quermassintegrals. J. Inequal. Appl. 2019, 85 (2019)

    Google Scholar 

  11. Ma, L.: A new proof of the Log-Brunn-Minkowski inequality. Geom. Dedicata 177, 75–82 (2015)

    MathSciNet  MATH  Google Scholar 

  12. Lv, S.-J.: The \(\varphi \)-Brunn–Minkowski inequality. Acta Math. Hung. 156, 226–239 (2018)

    MathSciNet  MATH  Google Scholar 

  13. Saroglou, C.: Remarks on the conjectured log-Brunn–Minkowski inequality. Geom. Dedicata 177, 353–365 (2015)

    MathSciNet  MATH  Google Scholar 

  14. Zhao, C.-J.: On the Orlicz–Brunn–Minkowski theory. Balkan J. Geom. Appl. 22, 98–121 (2017)

    MathSciNet  MATH  Google Scholar 

  15. Zhao, C.-J.: Inequalities for Orlicz mixed quermassintegrals. J. Convex Anal. 26(1), 129–151 (2019)

    MathSciNet  MATH  Google Scholar 

  16. Burago, Y.D., Zalgaller, V.A.: Geometric Inequalities. Springer, Berlin (1988)

    MATH  Google Scholar 

  17. Zhao, C.-J.: Orlicz–Aleksandrov–Fenchel inequality for Orlicz multiple mixed volumes. J. Funct. Spaces 2018, Article ID 9752178. https://doi.org/10.1155/2018/9752178

  18. Schneider, R.: Convex Bodies: The Brunn–Minkowski Theory. Cambridge University Press, Cambridge (1993)

    MATH  Google Scholar 

  19. Aleksandrov, A.D.: On the theory of mixed volumes. I. Extension of certain concepts in the theory of convex bodies. Mat. Sb. (N. S.) 2, 947–972 (1937)

    Google Scholar 

  20. Fenchel, W., Jessen, B.: Mengenfunktionen und konvexe Körper. Danske Vid. Selskab. Mat.-fys. Medd. 16, 1–31 (1938)

    MATH  Google Scholar 

  21. Busemann, H.: Convex Surfaces. Interscience, New York (1958)

    MATH  Google Scholar 

  22. Schneider, R.: Boundary Structure and Curvature of Convex Bodies, Contributions to Geometry, pp. 13–59. Birkhäuser, Basel (1979)

  23. Gardner, R.J.: Geometric Tomography, second edn. Cambridge University Press, New York (2006)

    MATH  Google Scholar 

  24. Lutwak, E.: The Brunn–Minkowski–Firey theory I. Mixed volumes and the Minkowski problem. J. Differ. Goem. 38, 131–150 (1993)

    MathSciNet  MATH  Google Scholar 

  25. Lutwak, E., Yang, D., Zhang, G.: \(L_{p}\) affine isoperimetric inequalities. J. Differ. Geom. 56, 111–132 (2000)

    MATH  Google Scholar 

  26. Lutwak, E., Yang, D., Zhang, G.: Sharp affine \(L_{p}\) Sobolev inequalities. J. Differ. Geom. 62, 17–38 (2002)

    MATH  Google Scholar 

  27. Firey, W.J.: \(p\)-means of convex bodies. Math. Scand. 10, 17–24 (1962)

    MathSciNet  MATH  Google Scholar 

  28. Firey, W.J.: Polar means of convex bodies and a dual to the Brunn–Minkowski theorem. Can. J. Math. 13, 444–453 (1961)

    MathSciNet  MATH  Google Scholar 

  29. Wang, X., Xu, W., Zhou, J.: Some logarithmic Minkowski inequalities for nonsymmetric convex bodies. Sci. China 60, 1857–1872 (2017)

    MathSciNet  MATH  Google Scholar 

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  1. Department of Mathematics, China Jiliang University, Hangzhou, 310018, People’s Republic of China

    Chang-Jian Zhao

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  1. Chang-Jian Zhao
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Correspondence to Chang-Jian Zhao.

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Research is supported by National Natural Science Foundation of China (11371334, 10971205).

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Zhao, CJ. The Log-Aleksandrov–Fenchel Inequality. Mediterr. J. Math. 17, 96 (2020). https://doi.org/10.1007/s00009-020-01521-7

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  • DOI: https://doi.org/10.1007/s00009-020-01521-7

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