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

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 integrability improving properties of the Heat semigroup associated to the Boltzmann measures µ α (dx) = (Z α ) −1 e −2|x| α dx, when α ∈ (1, 2). As an application we derive accurate isoperimetric inequalities for their products. This completes earlier works by Bobkov-Houdré and Talagrand, and provides a scale of dimension free isoperimetric inequalities as well as comparison theorems.

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This article investigates, by probabilistic methods, various geometric questions on B n p , the unit ball of ℓ n p . We propose realizations in terms of independent random variables of several distributions on B n p , including the normalized volume measure. These representations allow us to unify and extend the known results of the sub-independence of coordinate slabs in B n p . As another application, we compute moments of linear functionals on B n p , which gives sharp constants in Khinchine's inequalities on B n p and determines the ψ2-constant of all directions on B n p . We also study the extremal values of several Gaussian averages on sections of B n p (including mean width and ℓ-norm), and derive several monotonicity results as p varies. Applications to balancing vectors in ℓ2 and to covering numbers of polyhedra complete the exposition.

It is shown that if X 1 , X 2 , … X_1,X_2,\ldots are independent and identically distributed square-integrable random variables, then the entropy of the normalized sum \[ Ent ⁡ ( X 1 + ⋯ + X n n ) \operatorname {Ent} \left (\frac {X_{1}+\cdots + X_{n}}{\sqrt {n}} \right ) \] is an increasing function of n n . The result also has a version for non-identically distributed random variables or random vectors.