Dichotomies, structure, and concentration in normed spaces

Abstract: We use probabilistic, topological and combinatorial methods to establish the following deviation inequality: For any normed space $X=(\mathbb R^n ,|\cdot| )$ there exists an invertible linear map $T:\mathbb R^n \to \mathbb R^n$ with [ \mathbb P\left( \big| |TG| -\mathbb E|TG| \big| > \varepsilon \mathbb E|TG| \right) \leq C\exp \left( -c\max{ \varepsilon^2, \varepsilon } \log n \right),\quad \varepsilon>0, ] where $G$ is the standard $n$-dimensional Gaussian vector and $C,c>0$ are universal constants. It follows that for every $\varepsilon\in (0,1)$ and for every normed space $X=(\mathbb R^n,|\cdot|)$ there exists a $k$-dimensional subspace of $X$ which is $(1+\varepsilon)$-Euclidean and $k\geq c\varepsilon \log n/\log\frac{1}{\varepsilon}$. This improves by a logarithmic on $\varepsilon$ term the best previously known result due to G. Schechtman.