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

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 $L^1$-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.