Orlicz dual of log-Aleksandrov–Fenchel inequality

In this paper, we establish an Orlicz dual of the log-Aleksandrov–Fenchel inequality, by introducing two new concepts of dual mixed volume measures, and using the newly established Orlicz dual Aleksandrov–Fenchel inequality. The Orlicz dual log-Aleksandrov– Fenchel inequality in special cases yields the classical dual Aleksandrov–Fenchel inequality and some dual logarithmic Minkowski type inequalities, respectively. Moreover, the dual log-Aleksandrov–Fenchel inequality is therefore also derived.

Keywords:

dual mixed volume, Lp-dual mixed volume, Orlicz multiple dual mixed volume, dual logarithmic Minkowski inequality dual Aleksandrov–Fenchel inequality,

PDF

___

[1] K. Chou and X. Wang, A logarithmic Gauss curvature flow and the Minkowski problem, Ann. Inst. Henri Poincaré, Analyse non linéaire, 17 (6), 733-751, 2000.
[2] A. Colesanti and P. Cuoghi, The Brunn–Minkowski inequality for the n-dimensional logarithmic capacity of convex bodies, Potential Math. 22, 289-304, 2005.
[3] M. Fathi and B. Nelson, Free Stein kernels and an improvement of the free logarithmic Sobolev inequality , Adv. Math. 317, 193-223, 2017.
[4] W. J. Firey, Polar means of convex bodies and a dual to the Brunn–Minkowski theorem, Canad. J. Math. 13, 444-453, 1961.
[5] M. Henk and H. Pollehn, On the log-Minkowski inequality for simplices and parallelepipeds, Acta Math. Hungarica, 155, 141-157, 2018.
[6] S. Hou and J. Xiao, A mixed volumetry for the anisotropic logarithmic potential, J. Geom Anal. 28, 2018-2049, 2018.
[7] C. Li and W. Wang, Log-Minkowski inequalities for the $L_{p}$-mixed quermassintegrals, J. Inequal. Appl., 2019 (1), 2019.
[8] E. Lutwak, Dual mixed volumes, Pacific J. Math. 58, 531-538, 1975.
[9] E. Lutwak, Intersection bodies and dual mixed volumes, Adv. Math. 71 232-261, 1988.
[10] E. Lutwak, Centroid bodies and dual mixed volumes, Proc. London Math. Soc. 60, 365-391, 1990.
[11] S. Lv, The $\varphi$-Brunn–Minkowski inequality, Acta Math. Hungarica, 156, 226-239, 2018.
[12] L. Ma, A new proof of the log-Brunn–Minkowski inequality, Geom. Dedicata, 177, 75-82, 2015.
[13] C. Saroglou, Remarks on the conjectured log-Brunn–Minkowski inequality , Geom. Dedicata, 177, 353-365, 2015.
[14] R. Schneider, Convex Bodies: The Brunn–Minkowski Theory, Cambridge University Press, 1993.
[15] A. Stancu, The logarithmic Minkowski inequality for non-symmetric convex bodies, Adv. Appl. Math. 73, 43-58, 2016.
[16] W.Wang and M. Feng, The log-Minkowski inequalities for quermassintegrals, J. Math. Inequal. 11, 983995, 2017.
[17] W. Wang and G. Leng, $L_{p}$-dual mixed quermassintegrals, Indian J. Pure Appl. Math. 36, 177-188, 2005.
[18] W.Wang and L. Liu, The dual log-Brunn–Minkowski inequalities, Taiwanese J. Math. 20, 909-919, 2016.
[19] X. Wang, W. Xu and J. Zhou, Some logarithmic Minkowski inequalities for nonsymmetric convex bodies, Sci. China, 60, 1857-1872, 2017.
[20] C.-J. Zhao, The dual logarithmic Aleksandrov-Fenchel inequality, Balkan J. Geom. Appl. 25, 157-169, 2020.
[21] C.-J. Zhao, The log-Aleksandrov-Fenchel inequality, Mediterr J. Math. 17 (3), 1-14, 2020.
[22] C.-J. Zhao, The dual Orlicz-Aleksandrov-Fenchel inequality, Mathematics, 8 2005, 2020.
[23] C.-J. Zhao, Orlicz dual logarithmic Minkowski inequality, Math Inequal. Appl. 24 (4), 1031-1040, 2021.
[24] G. Zhu, The logarithmic Minkowski problem for polytopes, Adv. Math. 262, 909-931, 2014.