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On Dvoretzky's theorem for subspaces of Lp

Author:
Grigoris Paouris  Petros Valettas  


Journal:
Journal of Functional Analysis


Issue Date:
2018


Abstract(summary):

Abstract 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: P ( | ‖ Z ‖ − E ‖ Z ‖ | > ε E ‖ Z ‖ ) ≤ C exp ⁡ ( − cmin ⁡ { α p ε 2 n , ( ε n ) 2 / p } ) , 0 < ε < 1 , where Z is the standard n -dimensional Gaussian vector, 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 on the dimension of random almost spherical sections for these spaces. In particular, for any 2 < p < ∞ and every n -dimensional subspace X of L p , the Euclidean space ℓ 2 k can be ( 1 + ε ) -embedded into X with k ≥ cp min ⁡ { ε 2 n , ( ε n ) 2 / p } , where cp > 0 is a constant depending only on p . This improves upon the previously known estimate due to Figiel, Lindenstrauss and Milman.


Page:
2225-2225


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