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We establish an improved form of the classical logarithmic Sobolev inequality for the Gaussian measure restricted to probability densities which satisfy a Poincar\'e inequality.
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M. Ledoux. Logarithmic Sobolev inequalities for unbounded spin systems revisited. Séminaire de Probabilités XXXV, Lecture Notes in Math
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D. Cordero-Erausquin. Some applications of mass transport to Gaussian type inequalities. Arch. Rational Mech. Anal
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D. Bakry, F. Barthe, P. Cattiaux, A. Guillin. A simple proof of the Poincaré inequality in a large class of probability measures including log-concave cases. Elec. Comm. Prob
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N. Fusco, F. Maggi, A. Pratelli. The sharp quantitative isoperimetric inequality. Ann. of Math
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S. Bobkov, N. Gozlan, C. Roberto, P.-M. Samson. Bounds on the deficit in the logarithmic Sobolev inequality. J. Funct. Anal
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2013
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A. Figalli, F. Maggi, A. Pratelli. Sharp stability theorems for the anisotropic Sobolev and log-Sobolev inequalities on functions of bounded variation. Adv. Math
2013
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E. Indrei, D. Marcon. A quantitative log-Sobolev inequality for a two parameter family of functions. Int. Math. Res. Not. IMRN
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D. Bakry, I. Gentil, M. Ledoux. Analysis and geometry of Markov diffusion operators
2014
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A. Figalli, D. Jerison. Quantitative stability for the Brunn-Minkowski inequality. J. Eur. Math. Soc
2014
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