# 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} }

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…

## 2 Citations

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Let $\mathscr{C}_n=\{-1,1\}^n$ be the discrete hypercube equipped with the uniform probability measure $\sigma_n$. Talagrand's influence inequality (1994) asserts that there exists $C\in(0,\infty)$…

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## References

SHOWING 1-10 OF 68 REFERENCES

On the tightness of Gaussian concentration for convex functions

- MathematicsJournal 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

- MathematicsAdvances 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

- Mathematics
- 2017

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

- Mathematics
- 1994

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

- MathematicsSt. 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

- Mathematics
- 1995

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…