Brunn - Minkowski Inequality

– We present a one-dimensional version of the functional form of the geometric Brunn-Minkowski inequality in free (noncommutative) probability theory. The proof relies on matrix approximation as used recently by P. Biane and F. Hiai, D. Petz and Y. Ueda to establish free analogues of the logarithmic Sobolev and transportation cost inequalities for strictly convex potentials, that are recovered here from the Brunn-Minskowski inequality as in the classical case. The method is used to extend to the free setting the Otto-Villani theorem stating that the logarithmic Sobolev inequality implies the transportation cost inequality. It is used further to recover the free analogue of Shannon’s entropy power inequality put forward by S. Szarek and D. Voiculescu. 1. Classical Brunn-Minkowski and functional inequalities In its multiplicative form, the classical geometric Brunn-Minkowski inequality indicates that for all bounded Borel measurable sets A,B in R, and every θ ∈ (0, 1), vol ( θA+ (1− θ)B ) ≥ vol (A)vol (B)1−θ where θA+(1−θ)B = {θx+(1−θ)y;x ∈ A, y ∈ B} and where vol (·) denotes the volume element in R. Equivalently on functions (known as the Prékopa-Leindler theorem), whenever θ ∈ (0, 1) and u1, u2, u3 are non-negative measurable functions on R such that u3 ( θx+ (1− θ)y ) ≥ u1(x)u2(y) for all x, y ∈ R, (1) then ∫ u3dx ≥ (∫ u1dx )θ(∫ u2dx )1−θ (2) (cf. [Ga], [Ba2] for modern expositions). The Brunn-Minkowski inequality has been used recently in the investigation of functional inequalities for strictly log-concave densities such as logarithmic Sobolev or transportation cost inequalities (cf. [B-G-L], [Le1], [Le2], [Vi]...). Let dμ = e−Qdx be a probability measure on R such that, for some c > 0, Q(x)− c 2 |x| 2 is convex on R. Therefore, Q(θx+ (1− θ)y)− θQ(x)− (1− θ)Q(y) ≤ −cθ(1− θ) 2 |x− y| for all x, y ∈ R. The typical example is the standard Gaussian measure e−|x|2/2 dx (2π)n/2 (with c = 1). Let then f and g be two (bounded continuous) functions on R such that g(x) ≤ f(y) + c 2 |x − y| , x, y ∈ R. Choose u1 = e(1−θ)g−Q, u2 = e−θf−Q and u3 = e−Q satisfying thus (1). According to (2), for every θ ∈ (0, 1), log ∫ e(1−θ)gdμ+ 1− θ θ log ∫ e−θfdμ ≤ 0. When θ → 0, log ∫ edμ ≤ ∫ fdμ. (3) This inequality is actually the dual form of the quadratic transportation cost inequality W2(μ, ν) ≤ 1 c H(ν |μ) (4) holding for all probability measures ν on R, where W2 is the Wasserstein distance between probability measures and H (ν |μ) = ∫ log dν dμ dν is the relative entropy of ν << μ. The argument relies on the one side on the Monge-Kantorovitch-Rubinstein dual characterization of the Wasserstein metric as W2(μ, ν) = sup [ ∫ gdν − ∫ fdμ ] where the supremum runs over all (bounded continuous) functions f and g such that g(x) ≤ f(y)+ 1 2 |x−y| 2 for all x, y ∈ R (cf. e.g. [Vi]), and on the other on the entropic inequality ∫ gdν ≤ log ∫ edμ+H(ν |μ). (5) The Brunn-Minkowski theorem also covers the logarithmic Sobolev inequality for μ. Given ν <<μ, set f = log dν dμ (assumed to be smooth enough), and let gt, t > 0, be such that gt(x) ≤ f(y)+ 1 2t |x−y| , x, y ∈ R. Apply Brunn-Minkowski to u1 = e 1 θ gt−Q (t = 1−θ cθ ), u2 = e −Q, u3 = ef−Q, to get that