Low Correlation Noise Stability of Symmetric Sets

TL;DR

This paper investigates how symmetric sets in Euclidean space behave under Gaussian noise stability, revealing that simple symmetric shapes maximize stability in one dimension but not necessarily in higher dimensions.

Contribution

It provides the first known results showing both cases where symmetric sets maximize and do not maximize Gaussian noise stability, highlighting dimension-dependent behavior.

Findings

Intervals centered at the origin maximize stability in 1D

Balls do not always maximize stability in dimensions 2 and higher

Symmetric sets exhibit different stability properties depending on dimension

Abstract

We study the Gaussian noise stability of subsets A of Euclidean space satisfying A=-A. It is shown that an interval centered at the origin, or its complement, maximizes noise stability for small correlation, among symmetric subsets of the real line of fixed Gaussian measure. On the other hand, in dimension two and higher, the ball or its complement does not always maximize noise stability among symmetric sets of fixed Gaussian measure. In summary, we provide the first known positive and negative results for the Symmetric Gaussian Problem.