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.