Extremal properties of half-spaces for spherically invariant measures

Sudakov, V. N.; Tsirelson, Boris

doi:10.1007/bf01086099

Cited by 269 publications

(221 citation statements)

References 2 publications

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“…The prototypic result for all concentration theorems is arguably the Gaussian concentration inequality [14,62], which asserts that if G is a standard Gaussian vector in R n and f : R n → R is a 1-Lipschitz function, then for all t > 0,…”

Section: Introductionmentioning

confidence: 99%

Concentration inequalities for non-Lipschitz functions with bounded derivatives of higher order

Adamczak

Wolff

2014

Probab. Theory Relat. Fields

163

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Building on the inequalities for homogeneous tetrahedral polynomials in independent Gaussian variables due to R. Latała we provide a concentration inequality for not necessarily Lipschitz functions f : R n → R with bounded derivatives of higher orders, which holds when the underlying measure satisfies a family of Sobolev type inequalitiesSuch Sobolev type inequalities hold, e.g., if the underlying measure satisfies the logSobolev inequality (in which case C( p) ≤ C √ p) or the Poincaré inequality (then C( p) ≤ C p). Our concentration estimates are expressed in terms of tensor-product norms of the derivatives of f . When the underlying measure is Gaussian and f is a polynomial (not necessarily tetrahedral or homogeneous), our estimates can be reversed (up to a constant depending only on the degree of the polynomial). We also show that for polynomial functions, analogous estimates hold for arbitrary random vectors with independent sub-Gaussian coordinates. We apply our inequalities to general additive functionals of random vectors (in particular linear eigenvalue statistics of random matrices) and the problem of counting cycles of fixed length in Erdős-Rényi random graphs, obtaining new estimates, optimal in a certain range of parameters.

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Section: Introductionmentioning

confidence: 99%

Concentration inequalities for non-Lipschitz functions with bounded derivatives of higher order

Adamczak

Wolff

2014

Probab. Theory Relat. Fields

163

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“…the product measure with the density (2π) −n/2 e −|x| 2 /2 , where | · | is the Euclidean norm in R n . Borell [5] and Sudakov with Tsirelson [12] proved that in this case half-spaces {x : x, u ≥ λ} are extremal. As Bobkov and Houdré proved in [4], on the real line this result can be generalized into the case of an arbitrary symmetric log-concave measure.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…the exponential measure on the half-line, and h 1 is such that supp f ⊂ (h 1 , ∞). By Lemma 2, inequalities (12) and (13) will finish the proof of the theorem (we will see below that h 0 > 0 if a = 0), since (12) says that the hT neighbourhood of C below u is not less than the hT neighbourhood of the trapezoid between a and u (and similarly (13) gives us an analogous estimate above u, up to a term Lh 2 ).…”

Section: Then μ(C) = μ(A) and μ(C + Ht ) μ(A + Ht ) For Every H >mentioning

confidence: 96%

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Isoperimetric problem for exponential measure on the plane with $$\ell _1$$ ℓ 1 -metric

Strzelecka

2017

Positivity

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“…The answer to this question follows the development of isoperimetric theory: the surface-minimizing body in R n with prescribed Lebesgue measure is a ball [26,29,21]. In the sphere it is a cap, or geodesic ball [21] which in turn implies that in Gaussian space it is a half-space [2,27,4]. The answer to the noise stability question is analogous: half-spaces maximize the Gaussian noise stability among all sets of a given measure [3,23,12].…”

Section: Introductionmentioning

confidence: 99%

Standard simplices and pluralities are not the most noise stable

2016

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Abstract. The Standard Simplex Conjecture and the Plurality is Stablest Conjecture are two conjectures stating that certain partitions are optimal with respect to Gaussian and discrete noise stability respectively. These two conjectures are natural generalizations of the Gaussian noise stability result by Borell (1985) and the Majority is Stablest Theorem (2004). Here we show that the standard simplex is not the most stable partition in Gaussian space and that Plurality is not the most stable low influence partition in discrete space for every number of parts k ≥ 3, for every value ρ = 0 of the noise and for every prescribed measures for the different parts as long as they are not all equal to 1/k. Our results do not contradict the original statements of the Plurality is Stablest and Standard Simplex Conjectures in their original statements concerning partitions to sets of equal measure. However, they indicate that if these conjectures are true, their veracity and their proofs will crucially rely on assuming that the sets are of equal measures, in stark contrast to Borell's result, the Majority is Stablest Theorem and many other results in isoperimetric theory. Given our results it is natural to ask for (conjectured) partitions achieving the optimum noise stability.

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Extremal properties of half-spaces for spherically invariant measures

Cited by 269 publications

References 2 publications

Concentration inequalities for non-Lipschitz functions with bounded derivatives of higher order

Concentration inequalities for non-Lipschitz functions with bounded derivatives of higher order

Isoperimetric problem for exponential measure on the plane with $$\ell _1$$ ℓ 1 -metric

Standard simplices and pluralities are not the most noise stable

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