Robust dimension free isoperimetry in Gaussian space

Abstract

We prove the first robust dimension free isoperimetric result for the standard Gaussian measure ${\hat{I}}_n^3$ and the corresponding boundary measure $γ_n^+$ in ${\mathbb{R}}^n$. The main result in the theory of Gaussian isoperimetry (proven in the 1970s by Sudakov and Tsirelson, and independently by Borell) states that if ${\hat{I}}_n^3(A)=1/2$ then the surface area of $A$ is bounded by the surface area of a half-space with the same measure, $γ_n^+(A)\leq(2π)^{-1/2}$. Our results imply in particular that if $A\subset {\mathbb{R}}^n$ satisfies ${\hat{I}}_n^3(A)=1/2$ and $γ_n^+(A)\leq(2π)^{-1/2}+δ$ then there exists a half-space $B\subset {\mathbb{R}}^n$ such that ${\hat{I}}_n^3(A\hat{I}”B)\le C{\log }^{-1/2}(1/\hat{I}´)$ for an absolute constant $C$. Since the Gaussian isoperimetric result was established, only recently a robust version of the Gaussian isoperimetric result was obtained by Cianchi et al., who showed that ${\hat{I}}_n^3(A\hat{I}”B)\le C(n)\sqrt{\hat{I}}´$ for some function $C(n)$ with no effective bounds. Compared to the results of Cianchi et al., our results have optimal (i.e., no) dependence on the dimension, but worse dependence on $\hat{I}´$.

Published at http://dx.doi.org/10.1214/13-AOP860 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)