From Brunn-Minkowski to Brascamp-Lieb and to logarithmic Sobolev inequalities

We develop several applications of the Brunn-Minkowski inequality in the Pr ekopa-Leindler form. In particular, we show that an argument of B. Maurey m ay be adapted to deduce from the Prekopa-Leindler theorem the Brascamp-Lieb inequality for strictly convex potentials. We deduce similarly the logarit hmic Sobolev inequality for uniformly convex potentials for which we deal m ore generally with arbitrary norms and obtain some new results in this cont ext. Applications to transportation cost and to concentration on uniformly convex bodies complete the exposition.