Projection Body and Isoperimetric Inequalities for s-Concave Functions

Abstract

For a positive integer s, the projection body of an s-concave function $f:{\mathbb{R}^n} \to [0, + \infty )$, a convex body in the (n + s)-dimensional Euclidean space ${\mathbb{R}^{n + s}}$, is introduced. Associated inequalities for s-concave functions, such as, the functional isoperimetric inequality, the functional Petty projection inequality and the functional Loomis-Whitney inequality are obtained.

Keywords

Isoperimetric inequality / s-Concave functions / Projection body / The Petty projection inequality

References

[1] Alonso-Gutiérrez D. A reverse Rogers-Shephard inequality for log-concave functions. J. Geom. Anal., 2019, 29(1): 299-315

[2] Alonso-Gutiérrez D, Artstein-Avidan S, Merino B G Rogers-Shephard and local Loomis-Whitney type inequalities. Math. Ann., 2019, 374(3–4): 1719-1771

[3] Alonso-Gutiérrez D, Bernués J, Merino B G. An extension of Berwald’s inequality and its relation to Zhang’s inequality. J. Math. Anal. Appl., 2020, 486: 123875

[4] Alonso-Gutiérrez D, Bernués J, Merino B G. Zhang’s inequality for log-concave functions. GAFA Seminar Notes, 2020, 2256: 29-48

[5] Artstein-Avidan S, Klartag B, Milman V. The Santaló point of a function, and a functional form of Santaló inequality. Mathematika, 2004, 51: 33-48

[6] Artstein-Avidan S, Klartag B, Schütt C, Werner E. Functional affine-isoperimetry and an inverse logarithmic Sobolev inequality. J. Funct. Anal., 2012, 262: 4181-4204

[7] Artstein-Avidan S, Slomka B A. A note on Santaló inequality for the polarity transform and its reverse. Proc. Amer. Math. Soc., 2015, 143(4): 1693-1704

[8] Ball, K., Isometric problems in l _p and sections of convex sets, PhD thesis, Cambridge, 1986.

[9] Ball K. Volume ratios and a reverse isoperimetric inequality. J. London Math. Soc., 1991, 44: 351-359

[10] Barthe F, Böröczky K J, Fradelizi M. Stability of the functional forms of the Blaschke-Santaló inequality. Monatsh. Math., 2014, 173: 135-159

[11] Bolker E D. A class of convex bodies. Trans. Amer. Math. Soc., 1969, 145: 323-345

[12] Caglar U, Fradelizi M, Guedon O Functional versions of L _p-affine surface area and entropy inequalities. Int. Math. Res. Not., 2016, 2016(4): 1223-1250

[13] Caglar U, Werner E. Divergence for s-concave and log concave functions. Adv. Math., 2014, 257: 219-247

[14] Caglar U, Werner E. Mixed f-divergence and inequalities for log concave functions. Proc. London Math. Soc., 2015, 110(2): 271-290

[15] Cianchi A, Lutwak E, Yang D, Zhang G. Affine Moser-Trudinger and Morrey-Sobolev inequalities. Calc. Var. Partial Differ. Equ., 2009, 36: 419-436

[16] Colesanti A. Functional inequalities related to the Rogers-Shephard inequality. Mathematika, 2006, 53: 81-101

[17] Colesanti A, Fragalà I. The first variation of the total mass of log-concave functions and related inequalities. Adv. Math., 2013, 244: 708-749

[18] Fang N, Xu W, Zhou J, Zhu B. The sharp convex mixed Lorentz-Sobolev inequality. Adv. Appl. Math., 2019, 111: 101936 25pp

[19] Fang N, Zhou J. LYZ ellipsoid and Petty projection body for log-concave functions. Adv. Math., 2018, 340: 914-959

[20] Fang N, Zhou J. The Busemann-Petty problem on entropy of log-concave functions. Sci. China Math., 2022, 65: 2171-2182

[21] Fradelizi M, Meyer M. Some functional forms of Blaschke-Santaló inequality. Math. Z., 2007, 256: 379-395

[22] Fradelizi M, Meyer M. Some functional inverse Santaló inequality. Adv. Math., 2008, 218: 1430-1452

[23] Gardner R. The Brunn-Minkowski inequality. Bulletin (New Series) of the American Math. Soc., 2002, 39(3): 355-405

[24] Haberl C, Schuster F. Asymmetric affine L _p Sobolev inequalities. J. Funct. Anal., 2009, 257: 641-658

[25] 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

[26] Klartag B, Milman V. Geometry of log-concave functions and measures. Geom. Dedicata., 2005, 112: 169-182

[27] Lehec J. A simple proof of the functional Santaló inequality. C. R. Acad. Sci. Paris. Sér. I, 2009, 347: 55-58

[28] Lehec J. Partitions and functional Santaló inequalities. Arch. Math. (Basel), 2009, 92: 89-94

[29] Leindler L. On a certain converse of Hölder inequality II. Acta Sci. Math. (Szeged), 1972, 33: 217-223

[30] Lin Y. Affine Orlicz Pólya-Szegö principle for log-concave functions. J. Funct. Anal., 2017, 273: 3295-3326

[31] Loomis L H, Whitney H. An inequality related to isoperimetric inequality. Bull. Amer. Math. Soc., 1949, 55: 961-962

[32] Ludwig M, Xiao J, Zhang G. Sharp convex Lorentz-Sobolev inequalities. Math. Ann., 2011, 350: 169-197

[33] Lutwak E, Yang D, Zhang G. Sharp affine L _p Sobolev inequalities. J. Differ. Geom., 2002, 62: 17-38

[34] Milman V, Rotem L. α-concave functions and a functional extension of mixed volumes. Elctron. Res. Announc. Math. Sci., 2013, 20: 1-11

[35] Milman V, Rotem L. Mixed integral and related inequalities. J. Funct. Anal., 2013, 264: 570-604

[36] Petty, C., Projection bodies, Proc. Coll. Convexity, Copenhagen, 1965, Københavns Univ. Mat. Inst., 1967, 234–241.

[37] Petty, C., Isoperimetric problems, Proc. Conf. Convexty and Combinatorial Geometry (Univ. Oklahoma, 1971), Univ. Oklahoma, 1972, 26–41.

[38] Préekopa A. Logarithmic concave measures with applications to stochastic programming. Acta Sci. Math. (Szeged), 1971, 32: 335-343

[39] Préekopa A. On logarithmic concave measures and functions. Acta Sci. Math. (Szeged), 1973, 34: 335-343

[40] Rockafellar R T. Convex Analysis, 1970, Princeton: Princeton University Press

[41] Rotem L. Support functions and mean width for α-concave functions. Adv. Math., 2013, 243: 168-186

[42] Schneider R. Zu einem Problem von Shephard über die Projectionen konvexer Körper. Math. Z., 1967, 101: 71-82

[43] Wang T. The affine Sobolev-Zhang inequality on $BV({\mathbb{R}^n})$. Adv. Math., 2012, 230: 2457-2473

[44] Zhang G. Restricted chord projection and affine inequalities. Geom. Dedicata, 1991, 39: 213-222

[45] Zhang G. The affine Sobolev inequality. J. Differ. Geom., 1999, 53: 183-202