Between Paouris concentration inequality and variance conjecture
Annales de l'I.H.P. Probabilités et statistiques, Tome 46 (2010) no. 2, p. 299-312
Nous prouvons une inégalité inverse Hölder presque isométrique pour la norme euclidienne sur une boule d'Orlicz généralisée isotrope qui interpole l'inégalité de concentration de Paouris et la conjecture de la variance. Nous étudions dans ce sens le cas des corps convexes isotropes à base inconditionnelle et celui des corps convexes généraux.
We prove an almost isometric reverse Hölder inequality for the euclidean norm on an isotropic generalized Orlicz ball which interpolates Paouris concentration inequality and variance conjecture. We study in this direction the case of isotropic convex bodies with an unconditional basis and the case of general convex bodies.
@article{AIHPB_2010__46_2_299_0,
     author = {Fleury, B.},
     title = {Between Paouris concentration inequality and variance conjecture},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     publisher = {Gauthier-Villars},
     volume = {46},
     number = {2},
     year = {2010},
     pages = {299-312},
     doi = {10.1214/09-AIHP315},
     zbl = {1214.46006},
     mrnumber = {2667700},
     language = {en},
     url = {http://http://www.numdam.org/item/AIHPB_2010__46_2_299_0}
}
Fleury, B. Between Paouris concentration inequality and variance conjecture. Annales de l'I.H.P. Probabilités et statistiques, Tome 46 (2010) no. 2, pp. 299-312. doi : 10.1214/09-AIHP315. http://www.numdam.org/item/AIHPB_2010__46_2_299_0/

[1] M. Anttila, K. Ball and I. Perissinaki. The central limit problem for convex bodies. Trans. Amer. Math. Soc. 355 (2003) 4723-4735. | MR 1997580 | Zbl 1033.52003

[2] S. G. Bobkov. Remarks on the growth of Lp-norms of polynomials. In Geometric Aspects of Functionnal Analysis 27-35. Lecture Notes in Math. 1745. Springer, Berlin, 2000. | MR 1796711 | Zbl 0976.46005

[3] S. G. Bobkov. Spectral gap and concentration for some spherically symmetric probability measures. In Geometric Aspects of Functional Analysis - Israel Seminar 37-43. Lecture Notes in Math. 1807. Springer, Berlin, 2003. | MR 2083386 | Zbl 1052.60003

[4] S. G. Bobkov and A. Koldobsky. On the central limit property of convex convex bodies. In Geometric Aspects of Functional Analysis - Israel Seminar 44-52. Lecture Notes in Math. 1807. Springer, Berlin, 2003. | MR 2083387 | Zbl 1039.52003

[5] D. Cordero-Erausquin, M. Fradelizi and B. Maurey. The (B) conjecture for the Gaussian measure of dilates of symmetric convex sets and related problems. J. Funct. Anal. 214 (2004), 410-427. | MR 2083308 | Zbl 1073.60042

[6] B. Fleury, O. Guédon and G. Paouris. A stability result for mean width of Lp-centroid bodies. Adv. Math. 214 (2007) 865-877. | MR 2349721 | Zbl 1132.52012

[7] R. J. Gardner. The Brunn-Minkowski inequality. Bull. Amer. Math. Soc. 39 (2002) 355-405. | MR 1898210 | Zbl 1019.26008

[8] R. Latala and J. O. Wojtaszczyk. On the infimum convolution inequality. Available at arXiv: 0801.4036. | MR 2449135 | Zbl 1161.26010

[9] B. Klartag. Power-law estimates for the central limit theorem for convex sets. J. Funct. Anal. 245 (2007) 284-310. | MR 2311626 | Zbl 1140.52004

[10] B. Klartag. A Berry-Esseen type inequality for convex bodies with an unconditional basis. Probab. Theory Related Fields. 45 (2009) 1-33. | MR 2520120 | Zbl 1171.60322

[11] R. Kannan, L. Lovász and M. Simonovits. Isoperimetric problems for convex bodies and a localization lemma. Discrete Comput. Geom. 13 (1995) 541-559. | MR 1318794 | Zbl 0824.52012

[12] E. Milman. On the role of convexity in isoperimetry, spectral-gap and concentration. Available at arXiv: 0712.4092. | MR 2507637 | Zbl 1181.52008

[13] V. D. Milman and G. Schechtman. Asymptotic Theory of Finite Dimensional Normed Spaces. Lecture Notes in Math. 1200. Springer, Berlin, 1986. | MR 856576 | Zbl 0606.46013

[14] G. Paouris. Concentration of mass in convex bodies. Geom. Funct. Anal. 16 (2006) 1021-1049. | MR 2276533 | Zbl 1114.52004

[15] M. Pilipczuk and J. O. Wojtaszczyk. The negative association property for the absolute values of random variables equidistributed on a generalized Orlicz ball. Positivity 12 (2008) 421-474. | MR 2421144 | Zbl 1155.60006

[16] S. Sodin. An isoperimetric inequality on the lp balls. Ann. Inst. H. Poincaré Probab. Statist. 44 (2008) 362-373. | Numdam | MR 2446328 | Zbl 1181.60025

[17] J. O. Wojtaszczyk. The square negative correlation property for generalized Orlicz balls. In Geometric Aspects of Functional Analysis - Israel Seminar 305-313. Lecture Notes in Math. 1910. Springer, Berlin, 2007. | MR 2349615 | Zbl 1137.60007