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The Gaussian isoperimetric inequality and transportation. (English) Zbl 1033.60011
Summary: 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 \(\int \| \nabla f\| \{\log_+\| \nabla f\|\}^{1/2}\;W(x)dx < \infty\). This strengthens the quadratic logarithmic Sobolev inequality of L. Gross [ Am. J. Math 97, 1061–1083 (1976; Zbl 0318.46049)]. Let \(\mu(dx) = e^{-xi(x)}dx\) be a probability measure on \(\mathbb R^d\), where \(\xi\) is uniformly convex. M. Talagrand’s technique extends to monotone rearrangements in several dimensions [Geom. Funct. Anal. 6, 587–600 (1996; Zbl 0859.46030)], yielding a direct proof that \(\mu\) 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.

MSC:
60E15 Inequalities; stochastic orderings
58C35 Integration on manifolds; measures on manifolds
39B62 Functional inequalities, including subadditivity, convexity, etc.
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