Some Applications of Mass Transport to Gaussian-Type Inequalities

Cordero–Erausquin, Dario

doi:10.1007/s002050100185

Cited by 109 publications

(146 citation statements)

References 20 publications

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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%

“…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%

“…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. As explained in the section to follow, (26) can be viewed as what Villani [43] has called the above-tangent formulation of displacement convexity for the entropy.…”

Section: Theorem 7 (Bakry and Emerymentioning

confidence: 99%

“…We mention however that Bobkov, Gentil and Ledoux [7] showed that there exist many connections between transportation inequalities, Prékopa-Leindler inequalities and hypercontractivity of Hamilton-Jacobi equations. Also, it is easy to deduce Otto and Villani's "HWI"-type inequalities [35] from Theorem 8 as in [13].…”

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confidence: 95%

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Prékopa–Leindler type inequalities on Riemannian manifolds, Jacobi fields, and optimal transport

Cordero–Erausquin¹,

McCann

Schmuckenschläger

2006

Ann. Fac. Sci. Toulouse Math

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We investigate Prékopa-Leindler type inequalities on a Riemannian manifold M equipped with a measure with density e −V where the potential V and the Ricci curvature satisfy Hess x V + Ric x ≥ λ I for all x ∈ M , with some λ ∈ R. As in our earlier work [14], the argument uses optimal mass transport on M , but here, with a special emphasis on its connection with Jacobi fields. A key role will be played by the differential equation satisfied by the determinant of a matrix of Jacobi fields. We also present applications of the method to logarithmic Sobolev inequalities (the Bakry-Emery criterion will be recovered) and to transport inequalities. A study of the displacement convexity of the entropy functional completes the exposition. RésuméNousétudions l'extension d'inégalités de type Prékopa-Leindler au cas d'une variété riemannienne Méquipée d'une mesure ayant une densité e −V où le potentiel V et la courbure de Ricci vérifient Hess x V + Ric x ≥ λ I (∀x ∈ M ), pour un certain λ ∈ R. Nous ferons appel, comme dans notre travail précédent [14], au transport optimal de mesure. Mais nous exploiterons plus encore son lien avec les champs de Jacobi, ce qui permettra de ramener la discussionà l'étude du déterminant d'une matrice de champs de Jacobi. Nous présentonségalement d'autres applications de la méthode, en particulier aux inégalités de Sobolev logarithmiques (critère de BakryEmery) età l'étude de la convexité de déplacement de la fonctionnelle entropie.

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Section: The Displacement Convexity Point Of Viewsupporting

confidence: 53%

Section: Theorem 7 (Bakry and Emerymentioning

confidence: 99%

Section: Theorem 7 (Bakry and Emerymentioning

confidence: 99%

mentioning

confidence: 95%

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Prékopa–Leindler type inequalities on Riemannian manifolds, Jacobi fields, and optimal transport

Cordero–Erausquin¹,

McCann

Schmuckenschläger

2006

Ann. Fac. Sci. Toulouse Math

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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%

The Brunn-Minkowski inequality

Gardner

2002

Bull. Amer. Math. Soc.

749

561

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Abstract. In 1978, Osserman [124] wrote an extensive survey on the isoperimetric inequality. The Brunn-Minkowski inequality can be proved in a page, yet quickly yields the classical isoperimetric inequality for important classes of subsets of R n , and deserves to be better known. This guide explains the relationship between the Brunn-Minkowski inequality and other inequalities in geometry and analysis, and some applications.

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Push forward measures and concentration phenomena

Jimenez

Naszódi

Villa

2013

Mathematische Nachrichten

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Abstract. In this note we study how a concentration phenomenon can be transmitted from one measure µ to a push-forward measure ν. In the first part, we push forward µ by π :x K , and obtain a concentration inequality in terms of the medians of the given norms (with respect to µ) and the Banach-Mazur distance between them. This approach is finer than simply bounding the concentration of the push forward measure in terms of the BanachMazur distance between K and L. As a corollary we show that any normed probability space with good concentration is far from any high dimensional subspace of the cube. In the second part, two measures µ and ν are given, both related to the norm · L, obtaining a concentration inequality in which it is involved the Banach-Mazur distance between K and L and the Lipschitz constant of the map that pushes forward µ into ν. As an application, we obtain a concentration inequality for the cross polytope with respect to the normalized Lebesgue measure and the ℓ1 norm.

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Some Applications of Mass Transport to Gaussian-Type Inequalities

Cited by 109 publications

References 20 publications

Prékopa–Leindler type inequalities on Riemannian manifolds, Jacobi fields, and optimal transport

Prékopa–Leindler type inequalities on Riemannian manifolds, Jacobi fields, and optimal transport

The Brunn-Minkowski inequality

Push forward measures and concentration phenomena

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