The Gaussian Isoperimetric Inequality and Transportation

The Gaussian Isoperimetric Inequality and Transportation 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

The Gaussian Isoperimetric Inequality and Transportation

Positivity, Volume 7 (3) – Oct 4, 2004
22 pages

/lp/springer_journal/the-gaussian-isoperimetric-inequality-and-transportation-KYIAI0PNAR
Publisher
Springer Journals
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1023/A:1026242611940
Publisher site
See Article on Publisher Site

Abstract

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.

Journal

PositivitySpringer Journals

Published: Oct 4, 2004

References

• Polar factorization and monotone rearrangement of vector-valued functions
Brenier, Y.
• The regularity of mappings with a convex potential
Caffarelli, L.A.

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