Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-12T22:47:30.546Z Has data issue: false hasContentIssue false
  • English
  • Français

SPECTRAL GAP FOR SOME INVARIANT LOG-CONCAVE PROBABILITY MEASURES

Part of: Inequalities Distribution theory - Probability Classical measure theory

Published online by Cambridge University Press:  22 November 2010

Nolwen Huet
Nolwen Huet*
Affiliation:
Institut de Mathématiques de Toulouse, UMR CNRS 5219, Université de Toulouse, 31062 Toulouse, France (email: nolwen.huet@math.univ-toulouse.fr)
Get access

Abstract

We show that the conjecture of Kannan, Lovász, and Simonovits on isoperimetric properties of convex bodies and log-concave measures is true for log-concave measures of the form ρ(∣x∣B) dx on ℝn and ρ(t,∣x∣B) dx on ℝ1+n, where ∣x∣B is the norm associated to any convex body B already satisfying the conjecture. In particular, the conjecture holds for convex bodies of revolution.

MSC classification

Secondary: 28A75: Length, area, volume, other geometric measure theory 26D10: Inequalities involving derivatives and differential and integral operators 60E15: Inequalities; stochastic orderings
Type
Research Article
Information
Mathematika , Volume 57 , Issue 1 , January 2011 , pp. 51 - 62
Copyright
Copyright © University College London 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Barthe, F. and Wolff, P., Remarks on non-interacting conservative spin systems: the case of gamma distributions. Stochastic Process. Appl. 119(8) (2009), 2711–2723.Google Scholar
[2]Bobkov, S. G., Isoperimetric and analytic inequalities for log-concave probability measures. Ann. Probab. 27(4) (1999), 1903–1921.CrossRefGoogle Scholar
[3]Bobkov, S. G., Spectral gap and concentration for some spherically symmetric probability measures. In Geometric Aspects of Functional Analysis (Lecture Notes in Mathematics 1807), Springer (Berlin, 2003), 37–43.Google Scholar
[4]Bobkov, S. G., On isoperimetric constants for log-concave probability distributions. In Geometric Aspects of Functional Analysis (Lecture Notes in Mathematics 1910), Springer (Berlin, 2007), 81–88.CrossRefGoogle Scholar
[5]Bobkov, S. G. and Houdré, C., Isoperimetric constants for product probability measures. Ann. Probab. 25(1) (1997), 184–205.CrossRefGoogle Scholar
[6]Bollobás, B. and Leader, I., Edge-isoperimetric inequalities in the grid. Combinatorica 11(4) (1991), 299–314.CrossRefGoogle Scholar
[7]Borell, C., Convex measures on locally convex spaces. Ark. Mat. 12 (1974), 239–252.CrossRefGoogle Scholar
[8]Burago, Y. D. and Maz’ya, V. G., Potential Theory and Function Theory for Irregular Regions (Seminars in Mathematics 3, V. A. Steklov Mathematical Institute, Leningrad), Consultants Bureau (New York, 1969), translated from Russian.Google Scholar
[9]Cheeger, J., A lower bound for the smallest eigenvalue of the Laplacian. In Problems in Analysis (Papers Dedicated to Salomon Bochner, 1969), Princeton University Press (Princeton, NJ, 1970), 195–199.Google Scholar
[10]Hadwiger, H., Gitterperiodische Punktmengen und Isoperimetrie. Monatsh. Math. 76 (1972), 410–418.CrossRefGoogle Scholar
[11]Kannan, R., Lovász, L. and Simonovits, M., Isoperimetric problems for convex bodies and a localization lemma. Discrete Comput. Geom. 13(3–4) (1995), 541–559.CrossRefGoogle Scholar
[12]Klartag, B., Power-law estimates for the central limit theorem for convex sets. J. Funct. Anal. 245(1) (2007), 284–310.CrossRefGoogle Scholar
[13]Klartag, B., A Berry–Esseen type inequality for convex bodies with an unconditional basis. Probab. Theory Related Fields 145(1–2) (2009), 1–33.CrossRefGoogle Scholar
[14]Latała, R. and Wojtaszczyk, J. O., On the infimum convolution inequality. Studia Math. 189(2) (2008), 147–187.CrossRefGoogle Scholar
[15]Ledoux, M., Spectral gap, logarithmic Sobolev constant, and geometric bounds. In Surveys in Differential Geometry (Surveys in Differential Geometry, IX), International Press (Somerville, MA, 2004), 219–240.Google Scholar
[16]Leindler, L., On a certain converse of Hölder’s inequality. II. Acta Sci. Math. (Szeged) 33(3–4) (1972), 217–223.Google Scholar
[17]Maz’ja, V. G., The negative spectrum of the higher-dimensional Schrödinger operator. Dokl. Akad. Nauk SSSR 144 (1962), 721–722.Google Scholar
[18]Maz’ja, V. G., On the solvability of the Neumann problem. Dokl. Akad. Nauk SSSR 147 (1962), 294–296.Google Scholar
[19]Milman, E., On the role of convexity in isoperimetry, spectral gap and concentration. Invent. Math. 177(1) (2009), 1–43.CrossRefGoogle Scholar
[20]Milman, V. D. and Pajor, A., Isotropic position and inertia ellipsoids and zonoids of the unit ball of a normed n-dimensional space. In Geometric Aspects of Functional Analysis (1987–88) (Lecture Notes in Mathematics 1376), Springer (Berlin, 1989), 64–104.Google Scholar
[21]Prékopa, A., Logarithmic concave measures with application to stochastic programming. Acta Sci. Math. (Szeged) 32 (1971), 301–316.Google Scholar
[22]Prékopa, A., On logarithmic concave measures and functions. Acta Sci. Math. (Szeged) 34 (1973), 335–343.Google Scholar
[23]Rudin, W., Real and Complex Analysis, 3rd edn., McGraw-Hill Book Co. (New York, 1987).Google Scholar
[24]Sodin, S., An isoperimetric inequality on the ℓ p balls. Ann. Inst. H. Poincaré Probab. Statist. 44(2) (2008), 362–373.CrossRefGoogle Scholar