From concentration to logarithmic Sobolev and Poincaré inequalities

Abstract

We give a new proof of the fact that Gaussian concentration implies the logarithmic Sobolev inequality when the curvature is bounded from below, and also that exponential concentration implies Poincaré inequality under null curvature condition. Our proof holds on non-smooth structures, such as length spaces, and provides a universal control of the constants. We also give a new proof of the equivalence between dimension free Gaussian concentration and Talagrand's transport inequality.

From dimension free concentration to the Poincaré inequality

“…This happens for instance when α(r) = be −ar k , r ≥ 0, with k > 2 and a, b > 0. Theorem 1.6 completes a previous result obtained by the first author [17] (see also [20]), namely that the Gaussian dimension free concentration is characterized by a transport-entropy inequality. We now state this result and start by recalling some notation.…”

“…x ∈ X n (we should write Q p,(n) t , but we will omit, for simplicity, the superscripts p and (n) in the notation). In the next proposition, we recall a result from [20] that gives a new way to express concentration of measure (our first main tool). Proposition 1.16.…”

Kantorovich duality for general transport costs and applications

“…Below, we sketch the proof of Theorem 6.5, by revisiting and to some extent simplifying some of the arguments given in [54] with the help of the duality results developed in the present paper and in [24]. The first of these duality formulas is Kantorovich duality for the cost T given in Theorem 2.14.…”

Poincaré, modified logarithmic Sobolev and isoperimetric inequalities for Markov chains with non-negative Ricci curvature

“…The proof of Theorem 6.1 will again consist in first obtaining a weak version of the MLSI via the HWI inequality, and then tightening it. In the continuous setting, the corresponding result (and actually a much stronger one, as we shall discuss in the next section) was proven employing such a strategy in [15]. That work strongly relies on a self-tightening property of the logarithmic Sobolev inequality, which states that if a non-tight LSI of the form Ent π (f 2 ) ≤ cI(f ) + α holds and if α is small enough, then a tight LSI holds.…”