The Gaussian Isoperimetric Inequality and Transportation |
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Authors: | Blower Gordon |
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Affiliation: | (1) Department of Mathematics and Statistics, Lancaster University, Lancaster, LA1 4YF, UK |
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Abstract: | Any probability measure on d which satisfies the Gaussian or exponential isoperimetric inequality fulfils a transportation inequality for a suitable cost function. Suppose that W (x) dx satisfies the Gaussian isoperimetric inequality: then a probability density function f with respect to W (x) dx has finite entropy, provided that t. This strengthens the quadratic logarithmic Sobolev inequality of Gross (Amr. J. Math 97 (1975) 1061). Let (dx) = e–(x) dx be a probability measure on d, where is uniformly convex. Talagrand's technique extends to monotone rearrangements in several dimensions (Talagrand, Geometric Funct. Anal. 6 (1996) 587), yielding a direct proof that satisfies a quadratic transportation inequality. The class of probability measures that satisfy a quadratic transportation inequality is stable under multiplication by logarithmically bounded Lipschitz densities. |
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Keywords: | isoperimetric function transportation logarithmic Sobolev inequality Orlicz spaces |
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