Dichotomies, structure, and concentration in normed spaces

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 : (RRn)-R-n -> with p(|vertical bar vertical bar TG vertical bar vertical bar - E vertical bar vertical bar TG vertical bar vertical bar| > epsilon E vertical bar vertical bar TG vertical bar vertical bar) <= C exp (-cmax{epsilon(2), epsilon} log n), epsilon > 0, where G is the standard n-dimensional Gaussian vector and C, c > 0 are universal constants. It follows that for every epsilon is an element of (0, 1) and for every normed space X = (R-n, || . ||) there exists a k-dimensional subspace of X which is (1 + epsilon)-Euclidean and k >= c epsilon log n/log 1/epsilon. This improves by a logarithmic on epsilon term the best previously known result due to G. Schechtman. (C) 2018 Elsevier Inc. All rights reserved.