  For a large class of random matrices $A$ with i.i.d. entries we show that the
$\ell_1$-quotient property holds with probability exponentially close to 1. In
contrast to previous results, our analysis does not require concentration of
the entrywise distributions. We provide a unified proof that recovers
corresponding previous results for (sub-)Gaussian and Weibull distributions.
Our findings generalize known results on the geometry of random polytopes,
providing lower bounds on the size of the largest Euclidean ball contained in
the centrally symmetric polytope spanned by the columns of $A$. At the same
time, our results establish robustness of noise-blind $\ell_1$-decoders for
recovering sparse vectors $x$ from underdetermined, noisy linear measurements
$y = Ax + w$ under the weakest possible assumptions on the entrywise
distributions that allow for recovery with optimal sample complexity even in
the noiseless case. Our analysis predicts superior robustness behavior for
measurement matrices with super-Gaussian entries, which we confirm by numerical
experiments.
