Franck Barthe

On a reverse form of the Brascamp-Lieb inequality

Abstract. We prove a reverse form of the multidimensional Brascamp-Lieb inequality. Our method also gives a new way to derive the Brascamp-Lieb inequality and is rather convenient for the study of equality cases.

Solution of Shannon's problem on the monotonicity of entropy

It is shown that if X1, X2, . . . are independent and identically distributed square-integrable nrandom variables then the entropy of the normalized sum nEnt (X1+ · · · + Xn over √n) is an increasing function of n. This resolves an old problem which goes back to [6, 7, 5]. nThe result also has a version for non-identically distributed random variables or random vectors.

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 Mazjas capacity theory, and present equivalent forms relating the capacity of sets to their measure. Orlicz hypercontractivity efficiently describes the integrability improving properties of the Heat semigroup associated to the Boltzmann measuresnµa(dx) = (Za)-1 e-2|x|adx, when a I (1,2). As an application we derive accurate isoperimetric inequalities for their products. This completes earlier works by Bobkov-Houdre and Talagrand, and provides a scale of dimension free isoperimetric inequalities as well as comparison theorems

Entropy jumps in the presence of a spectral gap

It is shown that if X is a random variable whose density satisfies a Poincare inequality, and Y is an independent copy of X, then the entropy of (X + Y )/ p 2 is greater than that of X by a fixed fraction of the entropy gap between X and the Gaussian of the same variance. The argument uses a new formula for the Fisher information of a marginal, which can be viewed as a reverse form of the Brunn-Minkowski inequality (in its functional form due to Prekopa and Leindler).

Concentration for independent random variables with heavy tails

If a random variable is not exponentially integrable, it is known that no concentration inequality holds for an infinite sequence of independent copies. Under mild conditions, we establish concentration inequalities for finite sequences of

Combinatorial Optimization Over Two Random Point Sets

n

Inégalités de Brascamp-Lieb et convexité

independent copies, with good dependence in

The volume product of convex bodies with many hyperplane symmetries

n

A Short Solution to the Busemann-Petty Problem

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Transference Principles for Log-Sobolev and Spectral-Gap with Applications to Conservative Spin Systems

Let ((mathcal{X},mathcal{Y})) be a pair of random point sets in ({mathbb{R}}^{d}) of equal cardinal obtained by sampling independently 2n points from a common probability distribution μ. In this paper, we are interested by functions L of ((mathcal{X},mathcal{Y})) which appear in combinatorial optimization. Typical examples include the minimal length of a matching of (mathcal{X}) and (mathcal{Y}), the length of a traveling salesperson tour constrained to alternate between points of each set, or the minimal length of a connected bipartite r-regular graph with vertex set ((mathcal{X},mathcal{Y})). As the size n of the point sets goes to infinity, we give sufficient conditions on the function L and the probability measure μ which guarantee the convergence of (L(mathcal{X},mathcal{Y})) under a suitable scaling. In the case of the minimal length matching, we extend results of Dobric and Yukich, and Boutet de Monvel and Martin.