Interpolated inequalities between exponential and Gaussian, Orlicz hypercontractivity and isoperimetry

Barthe, Franck; Cattiaux, Patrick; Roberto, Cyril

doi:10.4171/rmi/482

Cited by 121 publications

(292 citation statements)

References 57 publications

(170 reference statements)

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“…Capacities were introduced in the 1960's by Maz'ya [60,61], Federer and Fleming [32], and were used by Bobkov and Houdré in [20,19]. We follow a variation on the definition given in [63] (for general q), which was extended by Barthe, Cattiaux and Roberto (with q = 2) in [7] (after being introduced in [10]). We conform to the definition implicitly used by Sodin in [81] and Sodin and the author in [66].…”

Section: Stability Of the Isoperimetric Profilementioning

confidence: 99%

On the role of convexity in isoperimetry, spectral gap and concentration

2009

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We show that for convex domains in Euclidean space, Cheeger's isoperimetric inequality, spectral gap of the Neumann Laplacian, exponential concentration of Lipschitz functions, and the a-priori weakest requirement that Lipschitz functions have arbitrarily slow uniform tail-decay, are all quantitatively equivalent (to within universal constants, independent of the dimension). This substantially extends previous results of Maz'ya, Cheeger, GromovMilman, Buser and Ledoux. As an application, we conclude a sharp quantitative stability result for the spectral gap of convex domains under convex perturbations which preserve volume (up to constants) and under maps which are "on-average" Lipschitz. We also provide a new characterization (up to constants) of the spectral gap of a convex domain, as one over the square of the average distance from the "worst" subset having half the measure of the domain. In addition, we easily recover and extend many previously known lower bounds on the spectral gap of convex domains, due to Payne-Weinberger, Li-Yau, KannanLovász-Simonovits, Bobkov and Sodin. The proof involves estimates on the diffusion semigroup following Bakry-Ledoux and a result from Riemannian Geometry on the concavity of the isoperimetric profile. Our results extend to the more general setting of Riemannian manifolds with density which satisfy the CD(0, ∞) curvature-dimension condition of BakryEmery.

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Section: Stability Of the Isoperimetric Profilementioning

confidence: 99%

On the role of convexity in isoperimetry, spectral gap and concentration

2009

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“…Here again we are inspired by [13] (Well Method). It turns out that the Well Method can be generalized to other F -Sobolev inequalities (see [5]). …”

Section: Introduction and Frameworkmentioning

confidence: 99%

Hypercontractivity for perturbed diffusion semigroups

Cattiaux¹

2005

Annales De La Faculté Des Sciences De Toulouse Mathématiques

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Abstract. µ being a nonnegative measure satisfying some Log-Sobolev inequality, we give conditions on F for the Boltzmann measure ν = e −2F µ to also satisfy some Log-Sobolev inequality. This paper improves and completes the final section in [6]. A general sufficient condition and a general necessary condition are given and examples are explicitly studied.Résumé. µétant une mesure positive satisfaisant une inégalité de Sobolev logarithmique, nous donnons des conditions sur F pour que la mesure de Boltzmann ν = e −2F µ satisfasseégalement une telle inégalité (améliorant et complétant ainsi la dernière partie de [6]). Les conditions obtenues sont illustrées par des exemples.

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“…In the case of small entropy, we will use so-called Φ-Sobolev inequalities (even if our context is less general), see Chafaï [Cha04] for a comprehensive review, and Barthe-Cattiaux-Roberto [BCR04] for a general approach in the case of measure µ α .…”

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confidence: 99%

Modified logarithmic Sobolev inequalities and transportation inequalities

Gentil

Guillin

Miclo

2005

Probab. Theory Relat. Fields

113

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We present a class of modified logarithmic Sobolev inequality, interpolating between Poincaré and logarithmic Sobolev inequalities, suitable for measures of the type exp(−|x| α ) or more complex exp(−|x| α log β (2 + |x|)) (α ∈]1, 2[ and β ∈ R) which lead to new concentration inequalities. These modified inequalities share common properties with usual logarithmic Sobolev inequalities, as tensorisation or perturbation, and imply as well Poincaré inequality. We also study the link between these new modified logarithmic Sobolev inequalities and transportation inequalities.

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Interpolated inequalities between exponential and Gaussian, Orlicz hypercontractivity and isoperimetry

Cited by 121 publications

References 57 publications

On the role of convexity in isoperimetry, spectral gap and concentration

On the role of convexity in isoperimetry, spectral gap and concentration

Hypercontractivity for perturbed diffusion semigroups

Modified logarithmic Sobolev inequalities and transportation inequalities

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