Robust dimension free isoperimetry in Gaussian space

Abstract

We prove the first robust dimension free isoperimetric result for the standard Gaussian measure ${\gamma }_n$ and the corresponding boundary measure ${\gamma }_n^{+}$ 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 ${\gamma }_n(A)=1/2$ then the surface area of $A$ is bounded by the surface area of a half-space with the same measure, ${\gamma }_n^{+}(A)\le (2\pi {)}^{-1/2}$. Our results imply in particular that if $A\subset {\mathbb{R}}^n$ satisfies ${\gamma }_n(A)=1/2$ and ${\gamma }_n^{+}(A)\le (2\pi {)}^{-1/2}+\delta$ then there exists a half-space $B\subset {\mathbb{R}}^n$ such that ${\gamma }_n(A\Delta B)\le C{\log }^{-1/2}(1/\delta )$ 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 ${\gamma }_n(A\Delta B)\le C(n)\sqrt{\delta }$ 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 $\delta$.

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)

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