  Let $\Gamma$ be an $N\times n$ random matrix with independent entries and
such that in each row entries are i.i.d. Assume also that the entries are
symmetric, have unit variances, and satisfy a small ball probabilistic estimate
uniformly. We investigate properties of the corresponding random polytope
$\Gamma^* B_1^N$ in $\mathbb{R}$ (the absolute convex hull of rows of
$\Gamma$). In particular, we show that $$
  \Gamma B_1^N \supset b^{-1} \left( B_{\infty}^n \cap \sqrt{\ln (N/n)}\, B_2^n
\right). $$ where $b$ depends only on parameters in small ball inequality. This
extends results of \cite{LPRT} and recent results of \cite{KKR}. This inclusion
is equivalent to so-called $\ell_1$-quotient property and plays an important
role in compressive sensing (see \cite{KKR} and references therein).
