  Let $X$ be a symmetric, isotropic random vector in $\mathbb{R}^m$ and let
$X_1...,X_n$ be independent copies of $X$. We show that under mild assumptions
on $\|X\|_2$ (a suitable thin-shell bound) and on the tail-decay of the
marginals $\langle X,u\rangle$, the random matrix $A$, whose columns are
$X_i/\sqrt{m}$ exhibits a Gaussian-like behaviour in the following sense: for
an arbitrary subset of $T\subset \mathbb{R}^n$, the distortion $\sup_{t \in T}
| \|At\|_2^2 - \|t\|_2^2 |$ is almost the same as if $A$ were a Gaussian
matrix.
  A simple outcome of our result is that if $X$ is a symmetric, isotropic,
log-concave random vector and $n \leq m \leq c_1(\alpha)n^\alpha$ for some
$\alpha>1$, then with high probability, the extremal singular values of $A$
satisfy the optimal estimate: $1-c_2(\alpha) \sqrt{n/m} \leq \lambda_{\rm min}
\leq \lambda_{\rm max} \leq 1+c_2(\alpha) \sqrt{n/m}$.
