  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.
