  This paper considers compressed sensing matrices and neighborliness of a
centrally symmetric convex polytope generated by vectors $\pm X_1,...,\pm
X_N\in\R^n$, ($N\ge n$). We introduce a class of random sampling matrices and
show that they satisfy a restricted isometry property (RIP) with overwhelming
probability. In particular, we prove that matrices with i.i.d. centered and
variance 1 entries that satisfy uniformly a sub-exponential tail inequality
possess this property RIP with overwhelming probability. We show that such
"sensing" matrices are valid for the exact reconstruction process of $m$-sparse
vectors via $\ell_1$ minimization with $m\le Cn/\log^2 (cN/n)$. The class of
sampling matrices we study includes the case of matrices with columns that are
independent isotropic vectors with log-concave densities. We deduce that if
$K\subset \R^n$ is a convex body and $X_1,..., X_N\in K$ are i.i.d. random
vectors uniformly distributed on $K$, then, with overwhelming probability, the
symmetric convex hull of these points is an $m$-centrally-neighborly polytope
with $m\sim n/\log^2 (cN/n)$.
