  We show that any $n\times m$ matrix $A$ can be approximated in operator norm
by a submatrix with a number of columns of order the stable rank of $A$. This
improves on existing results by removing an extra logarithmic factor in the
size of the extracted matrix. Our proof uses the recent solution of the
Kadison-Singer problem. We also develop a sort of tensorization technique to
deal with constraint approximation problems. As an application, we provide a
sparsification result with equal weights and an optimal approximate John's
decomposition for non-symmetric convex bodies. This enables us to show that any
convex body in $\mathbb{R}^n$ is arbitrary close to another one having $O(n)$
contact points and fills the gap left in the literature after the results of
Rudelson and Srivastava by completely answering the problem. As a consequence,
we also show that the method developed by Gu\'edon, Gordon and Meyer to
establish the isomorphic Dvoretzky theorem yields to the best known result once
we inject our improvements.

Functional Analysis;Metric Geometry