• Corpus ID: 174801421

# Hypercontractivity, and Lower Deviation Estimates in Normed Spaces

@article{Paouris2019HypercontractivityAL,
title={Hypercontractivity, and Lower Deviation Estimates in Normed Spaces},
author={Grigoris Paouris and Konstantin E. Tikhomirov and Petros Valettas},
journal={arXiv: Functional Analysis},
year={2019}
}
• Published 7 June 2019
• Mathematics
• arXiv: Functional Analysis
We consider the problem of estimating probabilities of lower deviation $\mathbb P\{\|G\| \leqslant \delta \mathbb E\|G\|\}$ in normed spaces with respect to the Gaussian measure. These estimates occupy central role in the probabilistic study of high-dimensional structures. It has been confirmed in several concrete situations, using ad hoc methods, that lower deviations exhibit very different and more complex behavior than the corresponding upper estimates. A characteristic example of this…
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## References

SHOWING 1-10 OF 68 REFERENCES
On the tightness of Gaussian concentration for convex functions
• P. Valettas
• Mathematics
Journal d'Analyse Mathématique
• 2019
The concentration of measure phenomenon in Gauss' space states that every $L$-Lipschitz map $f$ on $\mathbb R^n$ satisfies \[ \gamma_{n} \left(\{ x : | f(x) - M_{f} | \geqslant t \} \right) \leqslant
Dichotomies, structure, and concentration in normed spaces
• Mathematics
Advances in Mathematics
• 2018
Abstract We use probabilistic, topological and combinatorial methods to establish the following deviation inequality: For any normed space X = ( R n , ‖ ⋅ ‖ ) there exists an invertible linear map T
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 Superconcentration, and randomized Dvoretzky's theorem for spaces with 1-unconditional bases 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. A Remark on the Median and the Expectation of Convex Functions of Gaussian Vectors Ten years ago A. Ehrhard published an important paper, [1], in which he proved that if γn is a gaussian measure on R n, Φ is the normal distribution function, i.e \(\Phi (t)=\frac{1}{\sqrt{2\pi}}\int A Gaussian small deviation inequality for convex functions • Mathematics • 2016 Let Z be an n-dimensional Gaussian vector and let f: \mathbb R^n \to \mathbb R be a convex function. We show that:$$\mathbb P \left( f(Z) \leq \mathbb E f(Z) -t\sqrt{ {\rm Var} f(Z)} \right)
The variance of the $\ell _p^n$-norm of the Gaussian vector, and Dvoretzky’s theorem
• Mathematics
St. Petersburg Mathematical Journal
• 2019
Let $n$ be a large integer, and let $G$ be the standard Gaussian vector in $R^n$. Paouris, Valettas and Zinn (2015) showed that for all $p\in[1,c\log n]$, the variance of the $\ell_p^n$--norm of $G$
Embedding of ℓk∞ and a Theorem of Alon and Milman
Consider normalized vectors (xi) i ≤n in a Banach space X and set \( Av\left\{{\left\| {\sum\limits_{i = 1}^n {{\varepsilon_i}{x_i}}} \right\|;{\varepsilon_i} = \pm 1} \right\},\,{w_n} = \sup
Extremal properties of half-spaces for spherically invariant measures
• Mathematics
• 1978
Below we shall establish certain extremal properties of half-spaces for spherically symmetrical and, in particular, Gaussian (including infinite-dimensional) measures: we also prove inequalities for
Small Ball Estimates for Quasi-Norms
• Mathematics
• 2014
This note contains two types of small ball estimates for random vectors in finite-dimensional spaces equipped with a quasi-norm. In the first part, we obtain bounds for the small ball probability of