Skip to main content
SpringerLink
Log in

Interpolating Thin-Shell and Sharp Large-Deviation Estimates for Lsotropic Log-Concave Measures

  • Published:
Geometric and Functional Analysis Aims and scope Submit manuscript

Abstract

Given an isotropic random vector X with log-concave density in Euclidean space \({\mathbb{R}^n}\) , we study the concentration properties of |X| on all scales, both above and below its expectation. We show in particular that

$$\begin{array}{l} \mathbb{P}\left ( \left | |X| - \sqrt{n} \right | \geq t\sqrt{n} \right ) \leq C \, {\rm exp} \left ( -cn^{1/2} {\rm min}(t^{3}, t) \right) \; \forall t \geq 0, \end{array}$$

for some universal constants c, C > 0. This improves the best known deviation results on the thin-shell and mesoscopic scales due to Fleury and Klartag, respectively, and recovers the sharp large-deviation estimate of Paouris. Another new feature of our estimate is that it improves when X is \({\psi_{\alpha}\, (\alpha \in(1, 2])}\) , in precise agreement with Paouris’ estimates. The upper bound on the thin-shell width \({\sqrt{\mathbb{V}{\rm ar}(|X|)}}\) we obtain is of the order of n 1/3, and improves down to n 1/4 when X is \({\psi_{2}}\) . Our estimates thus continuously interpolate between a new best known thin-shell estimate and the sharp large-deviation estimate of Paouris. As a consequence, a new best known bound on the Cheeger isoperimetric constant appearing in a conjecture of Kannan.Lovász-Simonovits is deduced.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Anttila M., Ball K., Perissinaki I.: The central limit problem for convex bodies, Trans. Amer. Math. Soc. 355(12), 4723–4735 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bakry D., Émery M.: Diffusions hypercontractives, in “Séminaire de probabilités, XIX, 1983/84”. Springer Lecture Notes in Math. 1123, 177–206 (1985)

    Article  Google Scholar 

  3. Ball K.: Logarithmically concave functions and sections of convex sets in \({\mathbb{R}^n}\) . Studia Math. 88(1), 69–84 (1988)

    MathSciNet  MATH  Google Scholar 

  4. Barlow R.E., Marshall A.W., Proschan F.: Properties of probability distributions with monotone hazard rate. Ann. Math. Statist. 34, 375–389 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  5. Berwald L.: Verallgemeinerung eines Mittelwertsatzes von. J. Favard für positive konkave Funktionen, Acta Math. 79, 17–37 (1947)

    MathSciNet  MATH  Google Scholar 

  6. Bobkov S.G.: On isoperimetric constants for log-concave probability distributions, Geometric Aspects of Functional Analysis, Israel Seminar 2004-2005. Springer Lecture Notes in Math. 1910, 81–88 (2007)

    Article  MathSciNet  Google Scholar 

  7. Bobkov S.G.: On concentration of distributions of random weighted sums. Ann. Probab. 31(1), 195–215 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bobkov S.G.: Spectral gap and concentration for some spherically symmetric probability measures, Geometric Aspects of Functional Analysis. Springer Lecture Notes in Math. 1807, 37–43 (2003)

    MathSciNet  Google Scholar 

  9. Bobkov S.G.: A. Koldobsky, On the central limit property of convex bodies, Geometric Aspects of Functional Analysis (2001-2002). Springer Lecture Notes in Math. 1807, 44–52 (2003)

    MathSciNet  Google Scholar 

  10. Bobkov S.G., Nazarov F.L.: On convex bodies and log-concave probability measures with unconditional basis, Geometric Aspects of Functional Analysis (2001-2002). Springer Lecture Notes in Math. 1807, 53–69 (2003)

    MathSciNet  Google Scholar 

  11. Borell C.: Complements of Lyapunov’s inequality. Math. Ann. 205, 323–331 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  12. Borell C.: Convex measures on locally convex spaces. Ark. Mat. 12, 239–252 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  13. Dafnis N., Paouris G.: Small ball probability estimates, ψ 2-behavior and the hyperplane conjecture. J. Funct. Anal. 258(6), 1933–1964 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  14. Fleury B.: Between Paouris concentration inequality and variance conjecture. Ann. Inst. Henri Poincaré Probab. Stat. 46(2), 299–312 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  15. Fleury B.: Concentration in a thin euclidean shell for log-concave measures. J. Func. Anal. 259, 832–841 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  16. Fleury B., Guédon O., Paouris G.: A stability result for mean width of lpcentroid bodies. Advances in Mathematics 214(2), 865–877 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  17. M. Fradelizi, Contributions à la géométrie des convexes.Méthodes fonctionnelles et probabilistes, Habilitation à Diriger des Recherches de l’Université Paris-Est Marne La Vallée, 2008; http://perso-math.univ-mlv.fr/users/fradelizi.matthieu/pdf/HDR.pdf

  18. Gardner R.J.: The Brunn–Minkowski inequality. Bull. Amer. Math. Soc. (N.S.) 39(3), 355–405 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  19. Grünbaum B.: Partitions of mass-distributions and of convex bodies by hyperplanes. Pacific J. Math. 10, 1257–1261 (1960)

    MathSciNet  MATH  Google Scholar 

  20. Haberl C.: L p intersection bodies. Adv. Math. 217(6), 2599–2624 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  21. Kannan R., Lovász L., Simonovits M.: Isoperimetric problems for convex bodies and a localization lemma. Discrete Comput. Geom. 13(3/4), 541–559 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  22. Klartag B.: On convex perturbations with a bounded isotropic constant. Geom. Funct. Anal. 16(6), 1274–1290 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  23. Klartag B.: A central limit theorem for convex sets. Invent. Math. 168, 91–131 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  24. Klartag B.: Power-law estimates for the central limit theorem for convex sets. J. Funct. Anal. 245, 284–310 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  25. Klartag B.: A Berry–Esseen type inequality for convex bodies with an unconditional basis. Probab. Theory Related Fields 45(1), 1–33 (2009)

    Article  MathSciNet  Google Scholar 

  26. Ledoux M.: The Concentration of Measure Phenomenon, Mathematical Surveys and Monographs 89. American Mathematical Society, Providence, RI (2001)

    Google Scholar 

  27. Lutwak E., Zhang G.: Blaschke-Santaló inequalities. J. Differential Geom. 47(1), 1–16 (1997)

    MathSciNet  MATH  Google Scholar 

  28. Milman E.: On gaussian marginals of uniformly convex bodies. J. Theoret. Prob. 22(1), 256–278 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  29. Milman E.: On the role of convexity in isoperimetry, spectral gap and concentration. Invent. Math. 177(1), 1–43 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  30. Milman V.D.: A new proof of A. Dvoretzky’s theorem on cross-sections of convex bodies. Funkcional. Anal. i Priložen. 5(4), 28–37 (1971)

    MathSciNet  Google Scholar 

  31. Milman V.D., Pajor A.: Isotropic position and inertia ellipsoids and zonoids of the unit ball of a normed n-dimensional space, Geometric Aspects of Functional Analysis (1987-1988). Springer Lecture Notes in Mathematics 1376, 64–104 (1989)

    Article  MathSciNet  Google Scholar 

  32. V.D. Milman, G. Schechtman, (With an appendix by M. Gromov), Asymptotic Theory of Finite-Dimensional Normed Spaces, Springer Lecture Notes in Mathematics1200 (1986).

  33. Paouris G.: ψ 2-estimates for linear functionals on zonoids. in “Geometric Aspects of Functional Analysis (2001-1002)”. Springer Lecture Notes in Mathematics 1807, 211–222 (2003)

    MathSciNet  Google Scholar 

  34. Paouris G.: Concentration of mass on convex bodies. Geom. Funct. Anal. 16(5), 1021–1049 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  35. G. Paouris, Small ball probability estimates for log-concave measures, Trans. Amer. Math. Soc., to appear.

Download references

Author information

Authors and Affiliations

  1. Laboratoire d’Analyse et de Mathématiques Appliquées, Université Paris-Est Marne La Vallée, Bd Descartes 5, Champs sur Marne, 77454, Marne La Vallée Cédex 2, France

    Olivier Guédon

  2. Department of Mathematics, Technion - Israel Institute of Technology, Haifa, 32000, Israel

    Emanuel Milman

Authors
  1. Olivier Guédon
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Emanuel Milman
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Olivier Guédon.

Additional information

E.M. supported by ISF and the Taub Foundation (Landau Fellow).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Guédon, O., Milman, E. Interpolating Thin-Shell and Sharp Large-Deviation Estimates for Lsotropic Log-Concave Measures. Geom. Funct. Anal. 21, 1043–1068 (2011). https://doi.org/10.1007/s00039-011-0136-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00039-011-0136-5

Keywords and phrases

2010 Mathematics Subject Classification

Search

Navigation