Low Correlation Noise Stability of Symmetric Sets

Heilman, Steven

doi:10.1007/s10959-020-01031-y

Cited by 5 publications

(6 citation statements)

References 26 publications

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“…This was observed in [Man17]. A similar observation was made in the context of noise stability in [Hei15]…”

Section: Introductionsupporting

confidence: 81%

Symmetric convex sets with minimal Gaussian surface area

Heilman¹

2021

American Journal of Mathematics

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Let Ω ⊆ R n+1 have minimal Gaussian surface area among all sets satisfying Ω = −Ω with fixed Gaussian volume. Let A = A x be the second fundamental form of ∂Ω at x, i.e., A is the matrix of first order partial derivatives of the unit normal vector at x ∈ ∂Ω. For any. Let A 2 be the sum of the squares of the entries of A, and let A 2→2 denote the 2 operator norm of A.It is shown that if Ω or Ω c is convex, and if eitherthen ∂Ω must be a round cylinder. That is, except for the case that the average value of A 2 is slightly less than 1, we resolve the convex case of a question of Barthe from 2001.The main tool is the Colding-Minicozzi theory for Gaussian minimal surfaces, which studies eigenfunctions of the Ornstein-Uhlenbeck type operator L = Δ − x, ∇ + A 2 + 1 associated to the surface ∂Ω. A key new ingredient is the use of a randomly chosen degree 2 polynomial in the second variation formula for the Gaussian surface area. Our actual results are a bit more general than the above statement. Also, some of our results hold without the assumption of convexity.

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“…This was observed in [Man17]. A similar observation was made in the context of noise stability in [Hei15]…”

Section: Introductionsupporting

confidence: 81%

Symmetric convex sets with minimal Gaussian surface area

Heilman¹

2021

American Journal of Mathematics

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“…However, it might still be the case that the solution of the problem is a cylinder B k r × R n−k , or its complement, for some k depending on the volume (see [22,Conjecture 1.3]). Here B k r denotes the k-dimensional ball with radius r. At least the results by Heilman [21,22] and La Manna [25] seem to indicate this.…”

Section: Introductionmentioning

confidence: 91%

“…We first observe that the claim (45) is almost trivial for i = 1. Indeed, by the Caccioppoli estimate (21) and by (36) from Lemma 2 (recall that we have chosen e (1) = v) we have…”

Section: Let Us Definementioning

confidence: 99%

“…The Gaussian isoperimetric problem for symmetric sets or its generalization to Gaussian noise is stated as an open problem in [8,18]. In the latter it was conjectured that the solution should be the ball or its complement, but this was recently disproved in [21]. Another natural candidate for the solution is the symmetric strip or its complement.…”

Section: Introductionmentioning

confidence: 99%

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Symmetry of minimizers of a Gaussian isoperimetric problem

Barchiesi

Julin

2019

Probab. Theory Relat. Fields

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We study an isoperimetric problem described by a functional that consists of the standard Gaussian perimeter and the norm of the barycenter. This second term has a repulsive effect, and it is in competition with the perimeter. Because of that, in general the solution is not the half-space. We characterize all the minimizers of this functional, when the volume is close to one, by proving that the minimizer is either the half-space or the symmetric strip, depending on the strength of the repulsive term. As a corollary, we obtain that the symmetric strip is the solution of the Gaussian isoperimetric problem among symmetric sets when the volume is close to one.2010 Mathematics Subject Class. 49Q20, 60E15.

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“…It was conjectured in Chakrabarti and Regev [34] and O'Donnell [130] that Pr(X n ∈ A, Y n ∈ B) is maximized by (B r , B s ) or by (B c r , B c s ) for some appropriate r, s > 0, where B r = {x n ∈ R n : x n 2 ≤ r} denotes the ball centered at the origin with radius r. This conjecture was, however, disproved by Heilman [83] in dimensions two and higher.…”

Section: Part Imentioning

confidence: 99%