Levels of concentration between exponential and Gaussian
Autour de l'inégalité de Brunn-Minkowski
Matchings and the variance of Lipschitz functions
We are interested in the rate function of the moderate deviation principle for the two-sample matching problem. This is related to the determination of 1-Lipschitz functions with maximal variance. We give an exact solution for random variables which have normal law, or are uniformly distributed on the Euclidean ball.
On Gaussian Brunn-Minkowski inequalities
We are interested in Gaussian versions of the classical Brunn-Minkowski inequality. We prove in a streamlined way a semigroup version of the Ehrhard inequality for m Borel or convex sets based on a previous work by Borell. Our method also yields semigroup proofs of the geometric Brascamp-Lieb inequality and of its reverse form, which follow exactly the same lines.
Interpolated inequalities between exponential and Gaussian, Orlicz hypercontractivity and isoperimetry.
We introduce and study a notion of Orlicz hypercontractive semigroups. We analyze their relations with general F-Sobolev inequalities, thus extending Gross hypercontractivity theory. We provide criteria for these Sobolev type inequalities and for related properties. In particular, we implement in the context of probability measures the ideas of Maz'ja's capacity theory, and present equivalent forms relating the capacity of sets to their measure. Orlicz hypercontractivity efficiently describes the...
Isoperimetry between exponential and Gaussian.
A simple proof of the Poincaré inequality for a large class of probability measures.
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