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The areas log-Minkowski inequality

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Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas Aims and scope Submit manuscript

Abstract

In this paper, we first propose and establish an Orlicz log-Minkowski inequality for affine surface areas by introducing new concepts of affine surface area measures and using the newly established Orlicz Minkowski inequality for mixed affine surface areas. The new Orlicz log-Minkowski inequality in special cases yields the classical Minkowski inequality for mixed affine surface areas, areas log-Minkowki inequality \(L_{p}\) log-Minkowski inequality for affine surface areas and other log-Minkowski type inequalities for affine surface areas.

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References

  1. Bonnesen, T., Fenchel, W.: Theorie der konvexen K\(\ddot{o}\)rper. Springer, Berlin (1934)

    Book  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Book  Google Scholar 

  4. Chou, K., Wang, X.: A logarithmic Gauss curvature flow and the Minkowski problem. Ann. Inst. Henri Poincaré, Analyse non linéaire, 17(6), 733–751 (2000)

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  7. He, B., Leng, G., Li, K.: Projection problems for symmetric polytopes. Adv. Math. 207, 73–90 (2006)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  11. Lutwak, E.: The Brunn-Minkowski-Firey Theory I: mixed volumes and the Minkowski problem. J. Differ. Geom. 38, 131–150 (1993)

    Article  MathSciNet  Google Scholar 

  12. Lutwak, E.: The Brunn–Minkowski–Firey theory II: affine and geominimal surface areas. Adv. Math. 118, 244–294 (1996)

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Book  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  21. Wang, W., Leng, G.: \(L_{p}\)-mixed affine surface area. J. Math. Anal. Appl. 335, 341–354 (2007)

    Article  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  25. Zhao, C.-J.: Orlicz affine surface area. Balkan J. Geom. Appl. 24, 100–118 (2019)

    MathSciNet  MATH  Google Scholar 

  26. Zhou, Y., He, B.: On LYZ’s conjecture for the \(U\)-functional. Adv. Appl. Math. 87, 43–57 (2017)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The author expresses his grateful thanks to the referee for his many excellent suggestions and comment.

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Authors and Affiliations

  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 areas log-Minkowski inequality. RACSAM 115, 131 (2021). https://doi.org/10.1007/s13398-021-01065-9

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  • DOI: https://doi.org/10.1007/s13398-021-01065-9

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