Any probability measure on $$\mathbb{R}$$ 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∥ $$\smallint \left\| {\nabla {\text{ }}f} \right\|\{ \log _{\text{ + }} \left\| {\nabla {\text{ }}f} \right\|\} ^{1/2} {\text{ }}W(x){\text{ d}}x < \infty $$ . This strengthens the quadratic logarithmic Sobolev inequality of Gross (Amr. J. Math 97 (1975) 1061). Let μ(dx) = e −ξ(x) dx be a probability measure on $$\mathbb{R}$$ d, where ξ 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 μ 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.
Positivity – Springer Journals
Published: Oct 4, 2004
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