  Finding the sparsest solution $\alpha$ for an under-determined linear system
of equations $D\alpha=s$ is of interest in many applications. This problem is
known to be NP-hard. Recent work studied conditions on the support size of
$\alpha$ that allow its recovery using L1-minimization, via the Basis Pursuit
algorithm. These conditions are often relying on a scalar property of $D$
called the mutual-coherence. In this work we introduce an alternative set of
features of an arbitrarily given $D$, called the "capacity sets". We show how
those could be used to analyze the performance of the basis pursuit, leading to
improved bounds and predictions of performance. Both theoretical and numerical
methods are presented, all using the capacity values, and shown to lead to
improved assessments of the basis pursuit success in finding the sparest
solution of $D\alpha=s$.
