The Annals of Probability

Robust dimension free isoperimetry in Gaussian space

Elchanan Mossel and Joe Neeman

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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, $\gamma_{n}^{+}(A)\leq(2\pi)^{-1/2}$. Our results imply in particular that if $A\subset\mathbb{R} ^{n}$ satisfies $\gamma_{n}(A)=1/2$ and $\gamma_{n}^{+}(A)\leq(2\pi)^{-1/2}+\delta$ then there exists a half-space $B\subset\mathbb{R} ^{n}$ such that $\gamma_{n}(A\Delta B)\leq C\smash{\log^{-1/2}}(1/\delta)$ 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 $\delta$.

Article information

Source
Ann. Probab., Volume 43, Number 3 (2015), 971-991.

Dates
First available in Project Euclid: 5 May 2015

Permanent link to this document
https://projecteuclid.org/euclid.aop/1430830275

Digital Object Identifier
doi:10.1214/13-AOP860

Mathematical Reviews number (MathSciNet)
MR3342656

Zentralblatt MATH identifier
1320.60063

Subjects
Primary: 60E15: Inequalities; stochastic orderings
Secondary: 26D10: Inequalities involving derivatives and differential and integral operators 68Q87: Probability in computer science (algorithm analysis, random structures, phase transitions, etc.) [See also 68W20, 68W40] 60G10: Stationary processes

Keywords
Noise stability Gaussian measure isoperimetric inequalities majority is stablest

Citation

Mossel, Elchanan; Neeman, Joe. Robust dimension free isoperimetry in Gaussian space. Ann. Probab. 43 (2015), no. 3, 971--991. doi:10.1214/13-AOP860. https://projecteuclid.org/euclid.aop/1430830275


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