We study some problems on exponential integrability, concentration of measure, and transportation cost related to logarithmic Sobolev inequalities. On the real line, we then give a characterization of those probablity measures which satisfy these inequalities.

ABSTRACT. We add two sections to [8] and answer some questions asked there. In the first section we give another derivation of Theorem 1.1 of [8], which reveals the relation between the entropy formula, (1.4) of [8], and the well-known Li–Yau’s gradient estimate. As a by-product we obtain the sharp estimates on ‘Nash’s entropy ’ for manifolds with nonnegative Ricci curvature. We also show that the equality holds in Li–Yau’s gradient estimate, for some positive solution to the heat equation, at some positive time, implies that the complete Riemannian manifold with nonnegative Ricci curvature is isometric to R n. In the second section we derive a dual entropy formula which, to some degree, connects Hamilton’s entropy with Perelman’s entropy in the case of Riemann surfaces. 1. The relation with Li–Yau’s gradient estimates In this section we provide another derivation of Theorem 1.1 of [8] and discuss its relation with Li–Yau’s gradient estimates on positive solutions of heat equation. The formulation gives a sharp upper and lower bound estimates on Nash’s ‘entropy quantity ’ − � M H log Hdvin the case M has nonnegative Ricci curvature, where H is the fundamental solution (heat kernel) of the heat equation. This section is following the ideas in the Section 5 of [9]. Let u(x, t) be a positive solution to � ∂ ∂t − � � u(x, t) = 0 with �

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. We show that the entropy functional exhibits a quasi{factorization property with respect to a pair of weakly dependent {algebras. As an application we give a simple proof that the Dobrushin and Shlosman's complete analyticity condition, for a Gibbs specication with nite range summable interaction, implies uniform logarithmic Sobolev inequalities. This result has been previously proven using several dierent techniques. The advantage of our approach is that it relies almost entirely on a general property of the entropy, while very little is assumed on the Dirichlet form. No topology is introduced on the single spin space, thus discrete and continuous spins can be treated in the same way. Key Words: Entropy, Logarithmic Sobolev inequalities, Gibbs measures Mathematics Subject Classication 2000: 82B20, 82C20, 39B62 v1.0 1. Introduction Logarithmic Sobolev inequalities have been introduced in [Gr1] where it has been shown that Z R d f 2 (x) log jf(x)j d (dx) Z R d jr...

We study in details the isoperimetric profile of product probability measures with tails between the exponential and the Gaussian regime. In particular we exhibit many examples where coordinate half-spaces are approximate solutions of the isoperimetric problem. 1

Introduction Motivation, Examples, Statements of Results It is well known that the Sobolev inequality Z R n jrf(x)jd¯(x) n! 1=n n `Z R n jf(x)j n n\Gamma1 d¯(x) 'n\Gamma1 n ; n 2; (1.1) where f is a compactly supported smooth function on the Euclidean space R n , where ¯ is the Lebesgue measure in R n , and where ! n is the volume of the unit ball in R n , is equivalent to the isoperimetric property of the balls in R n . This isoperimetric property

Brascamp–Lieb-type, weighted Poincaré-type and related analytic inequalities are studied for multidimensional Cauchy distributions and more general κ-concave probability measures (in the hierarchy of convex measures). In analogy with the limiting (infinitedimensional log-concave) Gaussian model, the weighted inequalities fully describe the measure concentration and large deviation properties of this family of measures. Cheeger-type isoperimetric inequalities are investigated similarly, giving rise to a common weight in the class of concave probability measures under consideration. 1. Introduction. The

Abstract. The Heisenberg group is one of the simplest sub-Riemannian settings in which we can define non-elliptic Hörmander type generators. We can then consider coercive inequalities associated to such generators. We prove that a certain class of nontrivial Gibbs measures with quadratic interaction potential on an infinite product of Heisenberg groups satisfy logarithmic Sobolev inequalities. 1.

Spectral clustering is based on the spectral relaxation of the normalized/ratio graph cut criterion. While the spectral relaxation is known to be loose, it has been shown recently that a non-linear eigenproblem yields a tight relaxation of the Cheeger cut. In this paper, we extend this result considerably by providing a characterization of all balanced graph cuts which allow for a tight relaxation. Although the resulting optimization problems are non-convex and non-smooth, we provide an efficient first-order scheme which scales to large graphs. Moreover, our approach comes with the quality guarantee that given any partition as initialization the algorithm either outputs a better partition or it stops immediately. 1

Using measure-capacity inequalities we study new functional inequalities, namely Lq-Poincaré inequalities Varµ(f q) 1/q ∫