Mass Transport and Variants of the Logarithmic Sobolev Inequality

By Franck Barthe, A. V. Kolesnikov. 2007

75 citations recorded by the source corpus.

We develop the optimal transportation approach to modified log-Sobolev inequalities and to isoperimetric inequalities. Various sufficient conditions for such inequalities are given. Some of them are new even in the classical log-Sobolev case. The idea behind many of these conditions is that measures with a non-convex potential may enjoy such functional inequalities provided they have a strong integrability property that balances the lack of convexity. In addition, several known criteria are recovered in a simple unified way by transportation methods and generalized to the Riemannian setting.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/f0425286d63963b4382f751435e47e290cf6b1b2

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In Applied Mathematics

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Citations in the corpus

Transport Inequalities. A Survey (2010, background): “Recently Barthe and Kolesnikov have generalized Wang’s theorem to different functional inequalities and other convexity defects [8].”
On the role of convexity in isoperimetry, spectral gap and concentration (2007, methodology): “Bobkov’s approach was extended by Barthe [6] and subsequently by Barthe and Kolesnikov [8].”
Lyapunov conditions for Super Poincaré inequalities (2009, background): “See [21, 6] for an attempt involving modified log-Sobolev inequalities introduced in [19] and mass transport.”
Sharp Isoperimetric Inequalities and Model Spaces for Curvature-Dimension-Diameter Condition (2011, background): “[75, 16, 8, 9, 58]) considered an integrability condition of the form ∫”
Pointwise symmetrization inequalities for Sobolev functions and applications (2009, background): “This result was extended in [12] to the setting of Riemannian manifolds under appropriate curvature conditions.”
Functional affine-isoperimetry and an inverse logarithmic Sobolev inequality (2011, background): “There is a vast amount of literature on logarithmic Sobolev inequalities and related topics, e.g. [2, 5, 6, 9, 11, 15, 21].”
Isoperimetric and Concentration Inequalities - Equivalence under Curvature Lower Bound (2009, methodology): “Barthe and Kolesnikov [5] have obtained the most general results in this spirit (see in particular [5, Theorem 7.2]), treating general p > 1 under the convexity assumptions and p ≥ 2 under the κ-semi-convexity ones, again, with dimension dependent constants in the conclusion.”
On the role of convexity in functional and isoperimetric inequalities (2008, methodology): “We refer the reader to the recent paper of Barthe and Kolesnikov [8] and the references therein for an account of several other types of functional inequalities.”