Reinforcement of an inequality due to Brascamp and Lieb

In this paper, we obtain a reinforcement of an inequality due to Brascamp and Lieb and a reinforcement of Poincaré's inequality for general logarithmical concave measures on Rd. The formula used in the proof is related to theorems concerning the integration of log-concave functions (such as results of Prékopa and of Ball, Barthe and Naor). We also obtain a lower bound for the variance of the same family of measures.