# Superconcentration, and randomized Dvoretzky's theorem for spaces with 1-unconditional bases

@article{Tikhomirov2017SuperconcentrationAR,
title={Superconcentration, and randomized Dvoretzky's theorem for spaces with 1-unconditional bases},
author={Konstantin E. Tikhomirov},
journal={Journal of Functional Analysis},
year={2017},
volume={274},
pages={121-151}
}
Abstract Let n be a sufficiently large natural number and let B be an origin-symmetric convex body in R n in the l-position, and such that the space ( R n , ‖ ⋅ ‖ B ) admits a 1-unconditional basis. Then for any e ∈ ( 0 , 1 / 2 ] , and for random c e log ⁡ n / log ⁡ 1 e -dimensional subspace E distributed according to the rotation-invariant (Haar) measure, the section B ∩ E is ( 1 + e ) -Euclidean with probability close to one. This shows that the “worst-case” dependence on e in the randomized…
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## References

SHOWING 1-10 OF 36 REFERENCES
Random version of Dvoretzky’s theorem in ℓpn
• Mathematics
• 2017
We study the dependence on e in the critical dimension k(n,p,e) for which one can find random sections of the lpn-ball which are (1+e)-spherical. We give lower (and upper) estimates for k(n,p,e) for
Embedding ofl∞k in finite dimensional Banach spaces
• Mathematics
• 1983
AbstractLetx1,x2, ...,xn ben unit vectors in a normed spaceX and defineMn=Ave{‖Σi=1nε1xi‖:ε1=±1}. We prove that there exists a setA⊂{1, ...,n} of cardinality \left| A \right| \geqq \left[ {\sqrt n
The Random Version of Dvoretzky's Theorem in l_{\infty}^n
We show that with “high probability” a section of the `∞ ball of dimension k ≤ cε log n (c > 0 a universal constant) is ε close to a multiple of the Euclidean ball in this section. We also show that,
Euclidean Sections of Convex Bodies
This is a somewhat expanded form of a 4h course given, with small variations, first at the educational workshop Probabilistic methods in geometry, Bedlewo, Poland, July 6–12, 2008 and a few weeks
On Dvoretzky's theorem for subspaces of L
• Mathematics
Journal of Functional Analysis
• 2018
We prove that for any $2 \varepsilon \mathbb E\|Z\| \right) \leq C \exp \left (- c \min \left\{ \alpha_p \varepsilon^2 n, (\varepsilon n)^{2/p} \right\} \right), \quad 0 0$ is a constant depending
On Russo's Approximate Zero-One Law
Consider the product measure μ p on {0, 1} n , when 0 (resp. 1) is given weight 1-p (resp. p). Consider a monotone subset A of {0, 1} n . We give a precise quantitative form to the following
Superconcentration and Related Topics
Preface.- 1.Introduction.- 2.Markov semigroups.- 3.Super concentration and chaos.- 4.Multiple valleys.- 5.Talagrand's method for proving super concentration.- 6.The spectral method for proving super
Two observations regarding embedding subsets of Euclidean spaces in normed spaces
Abstract This paper contains two results concerning linear embeddings of subsets of Euclidean space in low-dimensional normed spaces. The first is an improvement of the known dependence on ɛ in
Asymptotic Theory Of Finite Dimensional Normed Spaces
• Mathematics
• 1986
The Concentration of Measure Phenomenon in the Theory of Normed Spaces.- Preliminaries.- The Isoperimetric Inequality on Sn?1 and Some Consequences.- Finite Dimensional Normed Spaces, Preliminaries.-
Extremal problems and isotropic positions of convex bodies
• Mathematics
• 2000
LetK be a convex body in ℝn and letWi(K),i=1, …,n−1 be its quermassintegrals. We study minimization problems of the form min{Wi(TK)|T ∈ SLn} and show that bodies which appear as solutions of such