Isoperimetry for spherically symmetric log-concave probability measures

TL;DR

This paper establishes a dimension-free isoperimetric inequality for a class of spherically symmetric log-concave probability measures on rom unctions of the Euclidean norm, with implications for high-dimensional probability.

Contribution

It proves a new isoperimetric inequality for measures with densities proportional to unctions of the Euclidean norm, extending previous results to a broader class of measures.

Findings

The inequality applies to measures with unctions like unctions of the form or rom 1.

The inequality is dimension-free under mild conditions on unctions.

It covers measures with unctions such as or rom unctions of the Euclidean norm.

Abstract

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 $\alpha\ge1$. Under mild assumptions on $\phi$, the inequality is dimension-free if $\lambda$ is chosen such that the covariance of $\mu$ is the identity.