Modified log-Sobolev inequalities and isoperimetry

We find sufficient conditions for a probability measure $\mu$ to satisfy aninequality of the type $$ \int_{\R^d} f^2 F\Bigl(\frac{f^2}{\int_{\R^d} f^2 d\mu} \Bigr) d \mu \le C \int_{\R^d} f^2 c^{*}\Bigl(\frac{|\nabla f|}{|f|}\Bigr) d \mu + B \int_{\R^d} f^2 d \mu, $$ where $F$ is concave and $c$ (a costfunction) is convex. We show that under broad assumptions on $c$ and $F$ theabove inequality holds if for some $\delta>0$ and $\epsilon>0$ one has $$\int_{0}^{\epsilon} \Phi\Bigl(\delta c\Bigl[\frac{t F(\frac{1}{t})}{{\mathcalI}_{\mu}(t)} \Bigr] \Bigr) dt < \infty, $$ where ${\mathcal I}_{\mu}$ is theisoperimetric function of $\mu$ and $\Phi = (y F(y) -y)^{*}$. In a partial case$${\mathcal I}_{\mu}(t) \ge k t \phi ^{1-\frac{1}{\alpha}} (1/t), $$ where$\phi$ 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 inequalitiesinterpolating between the $F$-Sobolev and modified inequalities of log-Sobolevtype. A basic example is given by convex measures satisfying certainintegrability assumptions.