Dichotomies, structure, and concentration in normed spaces

@article{Paouris2018DichotomiesSA,
  title={Dichotomies, structure, and concentration in normed spaces},
  author={Grigoris Paouris and Petros Valettas},
  journal={Advances in Mathematics},
  year={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 : R n → R n with P ( | ‖ T G ‖ − E ‖ T G ‖ | > e E ‖ T G ‖ ) ≤ C exp ⁡ ( − c max ⁡ { e 2 , e } log ⁡ n ) , e > 0 , where G is the standard n-dimensional Gaussian vector and C , c > 0 are universal constants. It follows that for every e ∈ ( 0 , 1 ) and for every normed space X = ( R n , ‖ ⋅ ‖ ) there… 
TALAGRAND’S INFLUENCE INEQUALITY REVISITED DARIO CORDERO-ERAUSQUIN AND ALEXANDROS ESKENAZIS
  • 2020
Let Cn = {−1, 1} be the discrete hypercube equipped with the uniform probability measure σn. Talagrand’s influence inequality (1994), also known as the L1 − L2 inequality, asserts that there exists C
Hypercontractivity, and Lower Deviation Estimates in Normed Spaces
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
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
Talagrand's influence inequality revisited
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 $C\in(0,\infty)$
Random Euclidean embeddings in finite dimensional Lorentz spaces
Quantitative bounds for random embeddings of Rk into Lorentz sequence spaces are given, with improved dependence on ε.

References

SHOWING 1-10 OF 74 REFERENCES
Embedding ofl∞k in finite dimensional Banach spaces
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
Random version of Dvoretzky’s theorem in ℓpn
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
Absolute and Unconditional Convergence in Normed Linear Spaces.
  • A. Dvoretzky, C. Rogers
  • Mathematics, Medicine
    Proceedings of the National Academy of Sciences of the United States of America
  • 1950
TLDR
The method of proof yields not only Theorem 1 but also the following result, which is easy to give examples of such series in Hilbert space and similar examples have been given for all the usually encountered infinitely dimensional Banach spaces.
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
Best constants for Lipschitz embeddings of metric spaces into $c_0$
We answer a question of Aharoni by showing that every separable metric space can be Lipschitz 2-embedded into $c_0$ and this result is sharp; this improves earlier estimates of Aharoni, Assouad and
Asymptotic Theory Of Finite Dimensional Normed Spaces
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.-
25 O ct 2 01 5 Random version of Dvoretzky ’ s theorem in l np
We study the dependence on ε in the critical dimension k(n, p, ε) that one can find random sections of the lp-ball which are (1+ε)-spherical. For any fixed n we give lower estimates for k(n, p, ε)
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
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
...
1
2
3
4
5
...