Between Paouris concentration inequality and variance conjecture
Annales de l'I.H.P. Probabilités et statistiques, Volume 46 (2010) no. 2, pp. 299-312.

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.

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.

DOI: 10.1214/09-AIHP315
Classification: 46B07,  46B09
Keywords: concentration inequalities, convex bodies
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Fleury, B. Between Paouris concentration inequality and variance conjecture. Annales de l'I.H.P. Probabilités et statistiques, Volume 46 (2010) no. 2, pp. 299-312. doi : 10.1214/09-AIHP315. http://www.numdam.org/articles/10.1214/09-AIHP315/

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

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

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

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

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

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

[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 | Zbl

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

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

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

[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 | Zbl

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

[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 | Zbl

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