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Properties of isoperimetric, functional and Transport-Entropy inequalities via concentration
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  • Published: 11 November 2010

Properties of isoperimetric, functional and Transport-Entropy inequalities via concentration

  • Emanuel Milman1 

Probability Theory and Related Fields volume 152, pages 475–507 (2012)Cite this article

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Abstract

Various properties of isoperimetric, functional, Transport-Entropy and concentration inequalities are studied on a Riemannian manifold equipped with a measure, whose generalized Ricci curvature is bounded from below. First, stability of these inequalities with respect to perturbation of the measure is obtained. The extent of the perturbation is measured using several different distances between perturbed and original measure, such as a one-sided L ∞ bound on the ratio between their densities, Wasserstein distances, and Kullback–Leibler divergence. In particular, an extension of the Holley–Stroock perturbation lemma for the log-Sobolev inequality is obtained, and the dependence on the perturbation parameter is improved from linear to logarithmic. Second, the equivalence of Transport-Entropy inequalities with different cost-functions is verified, by obtaining a reverse Jensen type inequality. The main tool used is a previous precise result on the equivalence between concentration and isoperimetric inequalities in the described setting. Of independent interest is a new dimension independent characterization of Transport-Entropy inequalities with respect to the 1-Wasserstein distance, which does not assume any curvature lower bound.

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Authors and Affiliations

  1. Department of Mathematics, University of Toronto, 40 St. George Street, Toronto, ON, M5S 2E4, Canada

    Emanuel Milman

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  1. Emanuel Milman
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Correspondence to Emanuel Milman.

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Cite this article

Milman, E. Properties of isoperimetric, functional and Transport-Entropy inequalities via concentration. Probab. Theory Relat. Fields 152, 475–507 (2012). https://doi.org/10.1007/s00440-010-0328-1

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  • Received: 19 May 2010

  • Revised: 01 October 2010

  • Published: 11 November 2010

  • Issue Date: April 2012

  • DOI: https://doi.org/10.1007/s00440-010-0328-1

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Keywords

  • Isoperimetric inequality
  • Log-Sobolev inequality
  • Transport-Entropy inequality
  • Concentration inequality
  • Stability under perturbation
  • Wasserstein distance

Mathematics Subject Classification (2000)

  • 60E15
  • 46G12
  • 60B99
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