On the Geometry of Log-Concave Probability Measures with Bounded Log-Sobolev Constant

Stavrakakis, Pantelis; Valettas, Petros

doi:10.1007/978-1-4614-6406-8_17

Cited by 2 publications

(2 citation statements)

References 22 publications

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“…for all smooth (or locally Lipschitz) functions f : R n → R. The n-dimensional Gaussian measure satisfies the log-Sobolev inequality with ρ = 1 (see [18]). The next lemma is essentially from [34] (see also [25]). We provide a sketch of proof for reader's convenience.…”

Section: Logarithmic Sobolev Inequalitymentioning

confidence: 99%

On Dvoretzky's theorem for subspaces of L

Paouris

Valettas

2018

Journal of Functional Analysis

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We prove that for any 2 < p < ∞ and for every n-dimensional subspace X of L p , represented on R n , whose unit ball B X is in Lewis' position one has the following two-level Gaussian concentration inequality:where Z is a standard n-dimensional Gaussian vectors, α p > 0 is a constant depending only on p and C, c > 0 are absolute constants. As a consequence we show optimal lower bound for the dimension of almost spherical sections for these spaces. In particular, for any 2 < p < ∞ and every n-dimensionalwhere c p > 0 is a constant depending only on p. This improves upon the previously known estimate due to Figiel, Lindenstrauss and V. Milman. n n+k cannot exceed Ck(X) (see [15] for a recent development on this fact). The proof of [22] gave the estimate c(ε) ≥ cε 2 / log 1 ε and this was improved to c(ε) ≥ cε 2 by Gordon in [13] and later, adopting the methods of V. Milman, by Schechtman in [27]. This dependence is known to be optimal in the setting of the randomized Dvoretzky theorem; see [30]). The works of Schechtman in [29] and Tikhomirov in [36] established that the dependence on ε in the randomized Dvoretzky for ℓ n ∞ is of the order ε/ log 1 ε and this is best possible. Optimal bounds on c(ε) in the randomized Dvoretzky for ℓ n p , 1 ≤ p ≤ ∞ have recently been studied in [25]. As far as the dependence on ε in the "existential version" of Dvoretzky's theorem is concerned, Schechtman proved in [28] that one can always (1 + ε)-embed ℓ k 2 in any 1 Here and elsewhere in this paper c and C denote positive absolute constants.Grigoris Paouris:

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Section: Logarithmic Sobolev Inequalitymentioning

confidence: 99%

On Dvoretzky's theorem for subspaces of L

Paouris

Valettas

2018

Journal of Functional Analysis

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“…Proof. The proof of the first estimate is essentially contained in [31]. The second one is direct application of the first for p = 2.…”

Section: Functional Inequalities On Gauss' Spacementioning

confidence: 99%

Random version of Dvoretzky's theorem in $\ell_p^n$

Paouris,

Valettas,

Zinn

2015

Preprint

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We study the dependence on ε in the critical dimension k(n, p, ε) for which one can find random sections of the ℓ n p -ball which are (1 + ε)-spherical. We give lower (and upper) estimates for k(n, p, ε) for all eligible values p and ε as n → ∞, which agree with the sharp estimates for the extreme values p = 1 and p = ∞.Toward this end, we provide tight bounds for the Gaussian concentration of the ℓ p -norm.

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On the Geometry of Log-Concave Probability Measures with Bounded Log-Sobolev Constant

Cited by 2 publications

References 22 publications

On Dvoretzky's theorem for subspaces of L

On Dvoretzky's theorem for subspaces of L

Random version of Dvoretzky's theorem in $\ell_p^n$

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