Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Between Paouris concentration inequality and variance conjecture

B. Fleury

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Abstract

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.

Résumé

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.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 46, Number 2 (2010), 299-312.

Dates
First available in Project Euclid: 11 May 2010

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1273584125

Digital Object Identifier
doi:10.1214/09-AIHP315

Mathematical Reviews number (MathSciNet)
MR2667700

Zentralblatt MATH identifier
1214.46006

Subjects
Primary: 46B07: Local theory of Banach spaces 46B09: Probabilistic methods in Banach space theory [See also 60Bxx]

Keywords
Concentration inequalities Convex bodies

Citation

Fleury, B. Between Paouris concentration inequality and variance conjecture. Ann. Inst. H. Poincaré Probab. Statist. 46 (2010), no. 2, 299--312. doi:10.1214/09-AIHP315. https://projecteuclid.org/euclid.aihp/1273584125


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References

  • [1] M. Anttila, K. Ball and I. Perissinaki. The central limit problem for convex bodies. Trans. Amer. Math. Soc. 355 (2003) 4723–4735.
  • [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.
  • [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.
  • [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.
  • [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.
  • [6] B. Fleury, O. Guédon and G. Paouris. A stability result for mean width of Lp-centroid bodies. Adv. Math. 214 (2007) 865–877.
  • [7] R. J. Gardner. The Brunn–Minkowski inequality. Bull. Amer. Math. Soc. 39 (2002) 355–405.
  • [8] R. Latala and J. O. Wojtaszczyk. On the infimum convolution inequality. Available at arXiv: 0801.4036.
  • [9] B. Klartag. Power-law estimates for the central limit theorem for convex sets. J. Funct. Anal. 245 (2007) 284–310.
  • [10] B. Klartag. A Berry–Esseen type inequality for convex bodies with an unconditional basis. Probab. Theory Related Fields. 45 (2009) 1–33.
  • [11] R. Kannan, L. Lovász and M. Simonovits. Isoperimetric problems for convex bodies and a localization lemma. Discrete Comput. Geom. 13 (1995) 541–559.
  • [12] E. Milman. On the role of convexity in isoperimetry, spectral-gap and concentration. Available at arXiv: 0712.4092.
  • [13] V. D. Milman and G. Schechtman. Asymptotic Theory of Finite Dimensional Normed Spaces. Lecture Notes in Math. 1200. Springer, Berlin, 1986.
  • [14] G. Paouris. Concentration of mass in convex bodies. Geom. Funct. Anal. 16 (2006) 1021–1049.
  • [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.
  • [16] S. Sodin. An isoperimetric inequality on the lp balls. Ann. Inst. H. Poincaré Probab. Statist. 44 (2008) 362–373.
  • [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.