Dichotomies, structure, and concentration in normed spaces
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2018 Elsevier Inc. We use probabilistic, topological and combinatorial methods to establish the following deviation inequality: For any normed space X=(Rn,) there exists an invertible linear map T:RnRn with P(|TGETG|>ETG)Cexp(cmax{2,}logn),>0, where G is the standard n-dimensional Gaussian vector and C,c>0 are universal constants. It follows that for every (0,1) and for every normed space X=(Rn,) there exists a k-dimensional subspace of X which is (1+)-Euclidean and kclogn/log[Formula presented]. This improves by a logarithmic on term the best previously known result due to G. Schechtman.