2002
DOI: 10.1007/s002050100185
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Some Applications of Mass Transport to Gaussian-Type Inequalities

Abstract: As discovered by Brenier, mapping through a convex gradient gives the optimal transport in R n . In the present article, this map is used in the setting of Gaussianlike measures to derive an inequality linking entropy with mass displacement by a straightforward argument. As a consequence, logarithmic Sobolev and transport inequalities are recovered. Finally, a result of Caffarelli on the Brenier map is used to obtain Gaussian correlation inequalities.

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Cited by 109 publications
(146 citation statements)
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“…The form of this inequality already suggests the fundamental result (26) of [13] and the preceding section. To complete the identification, it remains only to find the directional derivative of E(u s ), or at least show dE(u s ) ds s=0 + ≥ ∇f 0 · ∇θ dµ.…”
Section: The Displacement Convexity Point Of Viewsupporting
confidence: 53%
See 3 more Smart Citations
“…The form of this inequality already suggests the fundamental result (26) of [13] and the preceding section. To complete the identification, it remains only to find the directional derivative of E(u s ), or at least show dE(u s ) ds s=0 + ≥ ∇f 0 · ∇θ dµ.…”
Section: The Displacement Convexity Point Of Viewsupporting
confidence: 53%
“…The technicalities involved in this approach prevented them from giving a rigorous proof in the Riemannian case. We would like on the contrary to give a simple and direct proof along the lines of [13] and using the same techniques as above. In fact, as in [13], we will prove the following result from which log-Sobolev and transport inequalities can be recovered.…”
Section: Theorem 7 (Bakry and Emerymentioning
confidence: 99%
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“…Bobkov and Ledoux [24] derive (58) from the Prékopa-Leindler inequality (the "Brascamp-Lieb" in the title of [24] refers not to (59) below but to a different inequality of Brascamp and Lieb proved in [34]). Cordero-Erausquin [39] proves (58) directly using the transport of mass idea from Section 8.…”
Section: Information Theory Physics and Logarithmic Sobolev Inequalmentioning
confidence: 94%