Bernoulli

  • Bernoulli
  • Volume 23, Number 1 (2017), 134-158.

Transport proofs of weighted Poincaré inequalities for log-concave distributions

Dario Cordero-Erausquin and Nathael Gozlan

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We prove, using optimal transport tools, weighted Poincaré inequalities for log-concave random vectors satisfying some centering conditions. We recover by this way similar results by Klartag and Barthe–Cordero-Erausquin for log-concave random vectors with symmetries. In addition, we prove that the variance conjecture is true for increments of log-concave martingales.

Article information

Source
Bernoulli, Volume 23, Number 1 (2017), 134-158.

Dates
Received: July 2014
Revised: May 2015
First available in Project Euclid: 27 September 2016

Permanent link to this document
https://projecteuclid.org/euclid.bj/1475001351

Digital Object Identifier
doi:10.3150/15-BEJ739

Mathematical Reviews number (MathSciNet)
MR3556769

Zentralblatt MATH identifier
1378.60044

Keywords
log-concave measures transport inequalities weighted Poincaré inequalities

Citation

Cordero-Erausquin, Dario; Gozlan, Nathael. Transport proofs of weighted Poincaré inequalities for log-concave distributions. Bernoulli 23 (2017), no. 1, 134--158. doi:10.3150/15-BEJ739. https://projecteuclid.org/euclid.bj/1475001351


Export citation

References

  • [1] Alonso-Gutiérrez, D. and Bastero, J. (2013). The variance conjecture on some polytopes. In Asymptotic Geometric Analysis. Fields Inst. Commun. 68 1–20. New York: Springer.
  • [2] Anttila, M., Ball, K. and Perissinaki, I. (2003). The central limit problem for convex bodies. Trans. Amer. Math. Soc. 355 4723–4735 (electronic).
  • [3] Barthe, F. (2010). Transportation techniques and Gaussian inequalities. In Optimal Transportation, Geometry and Functional Inequalities. CRM Series 11 1–44. Pisa: Ed. Norm.
  • [4] Barthe, F. and Cordero-Erausquin, D. (2013). Invariances in variance estimates. Proc. Lond. Math. Soc. (3) 106 33–64.
  • [5] Barthe, F. and Kolesnikov, A.V. (2008). Mass transport and variants of the logarithmic Sobolev inequality. J. Geom. Anal. 18 921–979.
  • [6] Bobkov, S.G. (1999). Isoperimetric and analytic inequalities for log-concave probability measures. Ann. Probab. 27 1903–1921.
  • [7] Bobkov, S.G. (2003). On concentration of distributions of random weighted sums. Ann. Probab. 31 195–215.
  • [8] Bobkov, S.G. (2007). Large deviations and isoperimetry over convex probability measures with heavy tails. Electron. J. Probab. 12 1072–1100.
  • [9] Bobkov, S.G., Gozlan, N., Roberto, C. and Samson, P.-M. (2014). Bounds on the deficit in the logarithmic Sobolev inequality. J. Funct. Anal. 267 4110–4138.
  • [10] Bobkov, S.G. and Houdré, C. (1997). Isoperimetric constants for product probability measures. Ann. Probab. 25 184–205.
  • [11] Borell, C. (1974). Convex measures on locally convex spaces. Ark. Mat. 12 239–252.
  • [12] Borell, C. (1975). Convex set functions in $d$-space. Period. Math. Hungar. 6 111–136.
  • [13] Cordero-Erausquin, D. (2002). Some applications of mass transport to Gaussian-type inequalities. Arch. Ration. Mech. Anal. 161 257–269.
  • [14] Eldan, R. (2013). Thin shell implies spectral gap up to polylog via a stochastic localization scheme. Geom. Funct. Anal. 23 532–569.
  • [15] Eldan, R. and Klartag, B. (2014). Dimensionality and the stability of the Brunn–Minkowski inequality. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 13 975–1007.
  • [16] Gozlan, N. and Léonard, C. (2010). Transport inequalities. A survey. Markov Process. Related Fields 16 635–736.
  • [17] Gozlan, N., Roberto, C. and Samson, P.-M. (2013). Characterization of Talagrand’s transport-entropy inequalities in metric spaces. Ann. Probab. 41 3112–3139.
  • [18] Guédon, O. and Milman, E. (2011). Interpolating thin-shell and sharp large-deviation estimates for isotropic log-concave measures. Geom. Funct. Anal. 21 1043–1068.
  • [19] Hiriart-Urruty, J.-B. and Lemaréchal, C. (1993). Convex Analysis and Minimization Algorithms. II. Grundlehren der Mathematischen Wissenschaften 306. Berlin: Springer.
  • [20] Kannan, R., Lovász, L. and Simonovits, M. (1995). Isoperimetric problems for convex bodies and a localization lemma. Discrete Comput. Geom. 13 541–559.
  • [21] Klartag, B. (2007). A central limit theorem for convex sets. Invent. Math. 168 91–131.
  • [22] Klartag, B. (2009). A Berry–Esseen type inequality for convex bodies with an unconditional basis. Probab. Theory Related Fields 145 1–33.
  • [23] Klartag, B. (2010). High-dimensional distributions with convexity properties. In European Congress of Mathematics 401–417. Zürich: Eur. Math. Soc.
  • [24] Klartag, B. (2013). Poincaré inequalities and moment maps. Ann. Fac. Sci. Toulouse Math. (6) 22 1–41.
  • [25] Klartag, B. (2014). Concentration of measures supported on the cube. Israel J. Math. 203 59–80.
  • [26] Knothe, H. (1957). Contributions to the theory of convex bodies. Michigan Math. J. 4 39–52.
  • [27] Ledoux, M. (2001). The Concentration of Measure Phenomenon. Mathematical Surveys and Monographs 89. Providence, RI: Amer. Math. Soc.
  • [28] Marton, K. (1986). A simple proof of the blowing-up lemma. IEEE Trans. Inform. Theory 32 445–446.
  • [29] McCann, R.J. (1997). A convexity principle for interacting gases. Adv. Math. 128 153–179.
  • [30] Otto, F. and Villani, C. (2000). Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality. J. Funct. Anal. 173 361–400.
  • [31] Rachev, S.T. and Rüschendorf, L. (1998). Mass Transportation Problems. Vol. I. Probability and Its Applications (New York). New York: Springer.
  • [32] Talagrand, M. (1996). Transportation cost for Gaussian and other product measures. Geom. Funct. Anal. 6 587–600.
  • [33] Villani, C. (2009). Optimal Transport: Old and New. Grundlehren der Mathematischen Wissenschaften 338. Berlin: Springer.