Blower, Gordon
(2003)
*Gaussian isoperimetric inequalities and transportation.*
Positivity, 7 (3).
pp. 203-224.
ISSN 1385-1292

## 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.