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This paper considers compressed sensing matrices and neighborliness of acentrally symmetric convex polytope generated by vectors $\pm X_1,...,\pmX_N\in\R^n$, ($N\ge n$). We introduce a class of random sampling matrices andshow that they satisfy a restricted isometry property (RIP) with overwhelmingprobability. In particular, we prove that matrices with i.i.d. centered andvariance 1 entries that satisfy uniformly a sub-exponential tail inequalitypossess this property RIP with overwhelming probability. We show that such"sensing" matrices are valid for the exact reconstruction process of $m$-sparsevectors via $\ell_1$ minimization with $m\le Cn/\log^2 (cN/n)$. The class ofsampling matrices we study includes the case of matrices with columns that areindependent 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. randomvectors uniformly distributed on $K$, then, with overwhelming probability, thesymmetric convex hull of these points is an $m$-centrally-neighborly polytopewith $m\sim n/\log^2 (cN/n)$.

Restricted isometry property of matrices with independent columns and neighborly polytopes by random sampling