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 1dimensional refinements: the former is the one obtained by K.M. Ball and K. Böröczky in [4] and involves an L-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. 2010MSC: 52A40, 26D10, 39B62.