We present a new logarithmic Sobolev inequality adapted to a log-concave measure on between the exponential and the Gaussian measure. More precisely, assume that is a symmetric convex function on satisfying for large enough and with . We prove that the probability measure on satisfies a modified and adapted logarithmic Sobolev inequality: there exist three constants such that for all smooth functions ,
where is the Legendre-Fenchel transform of .
Cite this article
Ivan Gentil, Arnaud Guillin, Laurent Miclo, Modified logarithmic Sobolev inequalities in null curvature. Rev. Mat. Iberoam. 23 (2007), no. 1, pp. 235–258DOI 10.4171/RMI/493