By Christian Houdre, Michel Ledoux, Emanuel Milman, Mario Milman

The quantity includes the court cases of the foreign workshop on focus, sensible Inequalities and Isoperimetry, held at Florida Atlantic college in Boca Raton, Florida, from October 29-November 1, 2009.

The interactions among focus, isoperimetry and practical inequalities have ended in many major advances in practical research and likelihood theory.

vital development has additionally taken position in combinatorics, geometry, harmonic research and mathematical physics, to call yet a number of fields, with contemporary new functions in random matrices and data idea. This publication should still entice graduate scholars and researchers attracted to the attention-grabbing interaction among research, chance, and geometry.

**Read or Download Concentration, Functional Inequalities and Isoperimetry: International Workshop on Concentration, Functional Inequalities and Isoperiometry, October ... Boca Ra PDF**

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**Extra resources for Concentration, Functional Inequalities and Isoperimetry: International Workshop on Concentration, Functional Inequalities and Isoperiometry, October ... Boca Ra**

**Example text**

Introduction Let us denote by Lp the Banach space Lp (Rn , dx) of measurable functions deﬁned on Rn whose p-th power is integrable with respect to Lebesgue measure dx. 1) f g r ≤ f p g q , 1 1 1 + = + 1, p q r 1 < p, q, r < +∞, for functions f ∈ Lp and g ∈ Lq , which implies that if two functions are in (possibly diﬀerent) Lp -spaces, then their convolution is contained in a third Lp -space. 1) is reversed when 0 < p, q, r < 1. 1) was an open problem. Eventually, Beckner [9] proved Young’s inequality with the best possible constant.

C] Caﬀarelli, L. A. Monotonicity properties of optimal transportation and the FKG and related inequalities. Comm. Math. , 214 (2000), pp. 547–563. , and Maurey, B. The (B) conjecture for the Gaussian measure of dilates of symmetric convex sets and related problems. J. Funct. Anal. 214 (2004), no. 2, 410–427. [G] Gu´ edon, O. Kahane-Khinchine type inequalities for negative exponents. Mathematika, 46 (1999), 165–173. [K] Klartag, B. On convex perturbations with a bounded isotropic constant. Geom.

2) N Xj . M −1 j=i i∈[M ] This inequality was proved by Artstein, Ball, Barthe and Naor [1], and was used by them to aﬃrmatively resolve the long-standing conjecture of monotonicity in Barron’s entropic central limit theorem [5]. 1) yields d = 1 and hence N (X1 + . . 3) N (Xj ), j∈[M ] which is the classical Shannon-Stam entropy power inequality [39, 40]. This is already a nontrivial and interesting inequality, implying (as implicitly contained in [40]) for instance the logarithmic Sobolev inequality for the Gaussian usually attributed to Gross [26].