Isoperimetry for spherically symmetric log-concave probability measures

We prove an isoperimetric inequality for probability measures $\mu$ on ${\mathbb{R}}^n$ with density proportional to $\exp (-\varphi (\lambda \mathrm{\mid }x\mathrm{\mid }))$, where $\mathrm{\mid }x\mathrm{\mid }$ is the euclidean norm on ${\mathbb{R}}^n$ and $\varphi$ is a non-decreasing convex function. It applies in particular when $\varphi (x)=x^{\alpha }$ with $\alpha \ge 1$. Under mild assumptions on $\varphi$, the inequality is dimension-free if $\lambda$ is chosen such that the covariance of $\mu$ is the identity.