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Journal of Convex Analysis 21 (2014), No. 1, 289--305
Copyright Heldermann Verlag 2014

Lower Bounds for the Prékopa-Leindler Deficit by Some Distances Modulo Translations

Dorin Bucur
Lab. de Mathématiques, Université de Savoie, Campus Scientifique, 73376 Le-Bourget-Du-Lac, France

Ilaria Fragalá
Dip. di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italy

We discuss some refinements of the classical Prékopa-Leindler inequality, which consist in the addition of an extra-term depending on a distance modulo translations. Our results hold true on suitable classes of functions of n variables. They are based upon two different kinds of 1-dimensional refinements: the former is the one obtained by K. M. Ball and K. Böröczky ["Stability of the Prékopa-Leindler inequality", Mathematika 56 (2010) 339-356] and involves an L1-type distance on log-concave functions, the latter is new and involves the transport map onto the Lebesgue measure. Starting from each of these 1-dimensional refinements, we obtain an n-dimensional counterpart by exploiting a generalized version of the Cramér-Wold Theorem.

Keywords: Functional inequalities, Cramer-Wold Theorem, log-concave functions, mass transportation.

MSC: 52A40, 26D10, 39B62

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