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Transportation techniques and Gaussian inequalities. (English) Zbl 1206.60011

Ambrosio, Luigi (ed.), Optimal transportation, geometry and functional inequalities. Lectures of the school on the theory of optimal transportation and its applications, Pisa, Italy, 2008. Pisa: Edizioni della Normale (ISBN 978-88-7642-373-4/pbk). Centro di Ricerca Matematica Ennio De Giorgi (CRM) Series (Nuova Serie) 11, 1-44 (2010).
This is a well written and rich review on important functional Gaussian inequalities and their relationships. Recent developments are included, most direct proofs are given, and some striking applications are provided, too.
So the main purpose is to establish that:
(i) Dimension free Gaussian concentration is equivalent to quadratic transportation cost inequalities (\((T_2)\) for short) (which state that squared Wasserstein distance \((W^2_2)\) between two probability measures is controlled by their relative entropy);
(ii) Logarithmic Sobolev inequalities ((L.S.) for short) imply the preceding inequalities ((L.S.) state that entropy is controlled by energy);
(iii) The converse of (ii) is false.
Proofs for (i) use arguments by Talagrand (tensorisation property of \((T_2)\)) and Gozlan (who uses large deviations techniques to derive \((T_2)\) from Gaussian concentration).
The proof of (ii) uses stability of (L.S.) under tensorisation and a simple argument by Herbot.
The proof of (iii) uses calculations by several authors yielding in some cases nearly optimal constants in (L.S.) and \((T_2)\).
Finally, some other related topics are also reviewed, also with proofs; namely: sufficient criteria by Bakry-Emery and Wang ensuring (L.S.), Otto-Villani Theorem, and two pretty results by Hargé on the Gaussian correlation conjecture.
For the entire collection see [Zbl 1186.46005].

MSC:

60D05 Geometric probability and stochastic geometry
52A22 Random convex sets and integral geometry (aspects of convex geometry)
60E15 Inequalities; stochastic orderings
52A40 Inequalities and extremum problems involving convexity in convex geometry
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