# 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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