Geodesic
Lower Bounds for the Prékopa-Leindler Deficit by Some Distances Modulo Translations
Journal of convex analysis, Tome 21 (2014) no. 1, pp. 289-305

Voir la notice de l'article provenant de la source Heldermann Verlag

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.
Classification : 52A40, 26D10, 39B62
Mots-clés : Functional inequalities, Cramer-Wold Theorem, log-concave functions, mass transportation 
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     author = {D. Bucur and I. Fragal\'a},
     title = {Lower {Bounds} for the {Pr\'ekopa-Leindler} {Deficit} by {Some} {Distances} {Modulo} {Translations}},
     journal = {Journal of convex analysis},
     pages = {289--305},
     publisher = {mathdoc},
     volume = {21},
     number = {1},
     year = {2014},
     url = {http://geodesic.mathdoc.fr/item/JCA_2014_21_1_JCA_2014_21_1_a15/}
}
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D. Bucur; I. Fragalá. Lower Bounds for the Prékopa-Leindler Deficit by Some Distances Modulo Translations. Journal of convex analysis, Tome 21 (2014) no. 1, pp. 289-305. http://geodesic.mathdoc.fr/item/JCA_2014_21_1_JCA_2014_21_1_a15/