Logarithmic Sobolev inequality for log-concave measure from Prékopa-Leindler inequality
Résumé
We develop in this paper an amelioration of the method given by S. Bobkov and M. Ledoux. We prove by Prékopa-Leindler Theorem an optimal modified logarithmic Sobolev inequality adapted for all log-concave measure on $\dR^n$. This inequality implies results proved by Bobkov and Ledoux, the Euclidean Logarithmic Sobolev inequality generalized in the last years and it also implies some convex logarithmic Sobolev inequalities for large entropy.