# Revista Matemática Iberoamericana

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**Volume 27, Issue 1, 2011, pp. 93–122**

**DOI: 10.4171/RMI/631**

Published online: 2011-04-30

Isoperimetry for spherically symmetric log-concave probability measures

Nolwen Huet^{[1]}(1) Université de Toulouse, France

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

*Keywords: *Isoperimetric inequalities, log-concave measures

Huet Nolwen: Isoperimetry for spherically symmetric log-concave probability measures. *Rev. Mat. Iberoam.* 27 (2011), 93-122. doi: 10.4171/RMI/631