Low Correlation Noise Stability of Symmetric Sets
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Low Correlation Noise Stability of Symmetric Sets Steven Heilman1 Received: 10 March 2020 / Revised: 4 July 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
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. Keywords Noise stability · Symmetric sets · Gaussian measure · Optimization · Calculus of variations Mathematics Subject Classification (2010) 60E15 · 49Q10 · 46N30
1 Introduction Gaussian noise stability is a well-studied topic with connections to geometry of minimal surfaces [7], hypercontractivity and invariance principles [33], isoperimetric inequalities [20,21,23,28,32,35], sharp Unique Games hardness results in theoretical computer science [23–25,33], social choice theory, learning theory [22,26,27] and communication complexity [8,37,38]. In applications, it is often desirable to maximize noise stability. A sample result is the following well-known theorem of Borell, which has recently been re-proven and strengthened in various ways: Theorem 1.1 [3,11,28,32] Among all subsets of Euclidean space Rn of fixed Gaussian measure, a half space maximizes noise stability (for positive correlation).
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Steven Heilman [email protected] Department of Mathematics, University of Southern California, Los Angeles, CA 90089-2532, USA
123
Journal of Theoretical Probability
Here, a half space is any set of points lying on one side of a hyperplane, and noise stability is defined in Definition 1.3. A well-known Corollary of Theorem 1.1 says: among all subsets of Euclidean space Rn of fixed Gaussian measure, a half space has minimal Gaussian surface area. This statement may be surprising if one has only seen the isoperimetric inequality for Lebesgue measure. The latter inequality says: among all subsets of Euclidean space Rn of fixed Lebesgue measure, a ball has minimal surface area. The present paper concerns a variant of Theorem 1.1 where we restrict attention to symmetric sets. We say a subset A of Rn is symmetric if A = −A. Such a variant of Theorem 1.1 is a conjecture. Conjecture 1 (Informal, [1,8,34]) Among all symmetric subsets of Rn of fixed Gaussian measure, the ball centered at the origin or its complement maximizes noise stability (for positive correlation). If Conjecture 1 were true, then a Corollary would be: among all symmetric subsets of Euclidean space Rn of fixed Gaussian measure, a ball centered at the origin or its complement has minimal Gaussian surface area. So, by restricting our attention to symmetric sets, the isoperimetric sets for the Gaussian measure and Lebesgue m
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