Modified log-Sobolev inequalities and isoperimetry

Abstract

We find sufficient conditions for a probability measure $\mu$ to satisfy an inequality of the type

where $F$ is concave and $c$ (a~cost function) is convex. We show that under broad assumptions on $c$ and $F$ the above inequality holds if for some $\delta>0$ and $\varepsilon>0$ one has

where ${\mathcal I}_{\mu}$ is the isoperimetric function of $\mu$ and $\Phi = (y F(y) -y)^{*}$. In a partial case

where $\varphi$ is a concave function growing not faster than $\log$, $k>0$, $1 < \alpha \le 2$ and $t \le 1/2$, we establish a family of tight inequalities interpolating between the $F$-Sobolev and modified inequalities of log-Sobolev type. A basic example is given by convex measures satisfying certain integrability assumptions.