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We establish new tail estimates for order statistics and for the Euclidean norms of projections of an isotropic log‐concave random vector. More generally, we prove tail estimates for the norms of projections of sums of independent log‐concave random vectors, and uniform versions of these in the form of tail estimates for operator norms of matrices and their sub‐matrices in the setting of a log‐concave ensemble. This is used to study a quantity  A   k ,  m   that controls uniformly the operator norm of the sub‐matrices with  k  rows and  m  columns of a matrix  A  with independent isotropic log‐concave random rows. We apply our tail estimates of  A   k ,  m   to the study of restricted isometry property that plays a major role in the compressive sensing theory.

Tail estimates for norms of sums of log‐concave random vectors