Xiao Li , Jiazu Zhou . A FUNCTIONAL ORLICZ BUSEMANN-PETTY CENTROID INEQUALITY FOR LOG-CONCAVE FUNCTIONS[J]. Acta mathematica scientia, Series B, 2025 , 45(1) : 52 -71 . DOI: 10.1007/s10473-025-0105-y

[1] Barchiesi M, Capriani G, Fusco N, Pisante G. Stability of Pólya-Szegö inequality for log-concave functions. J Funct Anal, 2014, 267: 2264-2297
[2] Besau F, Hack T, Pivovarov P, Schuster F. Spherical centroid bodies. Amer J Math, 2023, 145: 515-542
[3] Brock F, Solynin A. An approach to symmetrization via polarization. Trans Amer Math Soc, 2000, 352: 1759-1796
[4] Burchard A. Steiner symmetrization is continuous in $W^{1,p}$. Geom Funct Anal, 1997, 7: 823-860
[5] Capriani G M. The Steiner rearrangement in any codimension. Calc Var PDE, 2014, 49: 517-548
[6] Campi S, Gronchi P. The $L_p$-Busemann-Petty centroid inequality. Adv Math, 2002, 167: 128-141
[7] Chen F, Zhou J, Yang C. On the reverse Orlicz Busemann-Petty centroid inequality. Adv Appl Math, 2011, 47: 820-828
[8] Cianchi A, Fusco N. Steiner symmetric extremals in Pólya-Szegö type inequalities. Adv Math, 2006, 203: 673-728
[9] Cianchi A, Lutwak E, Yang D, Zhang G. Affine Moser-Trudinger and Morrey-Sobolev inequalities. Calc Var PDE, 2009, 36: 419-436
[10] Colesanti A, Fragalà I. The first variation of the total mass of log-concave functions and related inequalities. Adv Math, 2013, 244: 708-749
[11] Cordero-Erausquin D, Klartag B. Moment measure. J Funct Anal, 2015, 268: 3834-3866
[12] Gardner R, Hug D, Weil W. The Orlicz-Brunn-Minkowski theory: a general framework, additions, and inequalities. J Differ Geom, 2014, 97: 427-476
[13] Haberl C, Schuster F. Asymmetric affine $L_p$ Sobolev inequalities. J Funct Anal, 2009, 257: 641-658
[14] Haberl C, Schuster F, Xiao J. An asymmetric affine Pólay-Szegö principle. Math Ann, 2012, 352: 517-542
[15] Haddad J, Jiménez C H, Montenegro M. Sharp affine Sobolev type inequalities via the $L_p$ Busemann-Petty centroid inequality. J Funct Anal, 2016, 271: 454-473
[16] Haddad J, Jiménez C H, Silva L A. An $L_p$-Functional Busemann-Petty Centroid Inequality. Int Math Res Not, 2021, 10: 7947-7965
[17] Klartag B, Milman E. Centroid bodies and the logarithmic Laplace transform-a unified approach. J Funct Anal, 2012, 262: 10-34
[18] Leichtweiβ K. Affine Geometry of Convex Bodies. Heidelberg: Johann Ambrosius Barth, 1998
[19] Lin Y. Affine Orlicz Pólya-Szegö principle for log-concave functions. J Funct Anal, 2017, 273: 3295-3326
[20] Lutwak E. Centroid bodies and dual mixed volumes. Proc London Math Soc, 1990: 60: 365-391
[21] Lutwak E, Yang D, Zhang G. $L_p$ affine isoperimetric inequalities. J Differ Geom, 2000, 56: 111-132
[22] Lutwak E, Yang D, Zhang G. Sharp affine $L_p$ Sobolev inequalities. J Differ Geom, 2002, 62: 17-38
[23] Lutwak E, Yang D, Zhang G. Orlicz centroid bodies. J Differ Geom, 2010, 84: 365-387
[24] Lutwak E, Yang D, Zhang G. Orlicz projection bodies. Adv Math, 2010, 223: 220-242
[25] Lutwak E, Zhang G. Blaschke-Santaló inequalities. J Differ Geom, 1997, 45: 1-16
[26] Nguyen V. New approach to the affine Pólya-Szegö principle and the stability version of the affine Sobolev inequality. Adv Math, 2016, 302: 1080-1110
[27] Nguyen V. Orlicz-Lorentz centroid bodies. Adv Appl Math, 2018, 92: 99-121
[28] Paouris G, Werner E. Relative entropy of cone measures and $L_p$ centroid bodies. Proc London Math Soc, 2012, 104: 253-286
[29] Paouris G, Pivovarov P. A probabilistic take on isoperimetric-type inequalities. Adv Math, 2012, 230: 1402-1422
[30] Petty C. Centroid surfaces. Pacific J Math, 1961, 11: 1535-1547
[31] Rotem L. Support functions and mean width for $\alpha$-concave functions. Adv Math, 2013, 243: 168-186
[32] Schneider R. Convex Bodies: The Brunn-Minkowski Theory. Cambridge: Cambridge University Press, 2014
[33] Volčcič A. Random Steiner symmetrizations of sets and functions. Calc Var PDE, 2013, 46: 555-569
[34] Werner E, Ye D. New $L_p$ affine isoperimetric inequalities. Adv Math, 2008, 218: 762-780
[35] Wu D, Zhou J. The LYZ centroid conjecture for star bodies. Sci China Math, 2018, 61: 1273-1286
[36] Zhang G. Centroid bodies and dual mixed volumes. Trans Amer Math Soc, 1994, 345: 777-801
[37] Zhang G. The affine Sobolev inequality. J Differ Geom, 1999, 53: 183-202
[38] Zhu G. The Orlicz centroid inequality for star bodies. Adv Appl Math, 2012, 48: 432-445