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

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