Restricted invertibility and the banach-mazur distance to the cube

We prove a normalized version of the restricted invertibility principle obtained by Spielman and Srivastava in [An elementary proof of the restricted invertibility theorem. Israel J. Math. 190 (2012), 83-91]. Applying this result, we get a new proof of the proportional Dvoretzky-Rogers factorization theorem recovering the best current estimate in the symmetric setting while we improve the best known result in the non-symmetric case. As a consequence, we slightly improve the estimate for the Banach-Mazur distance to the cube: the distance of every n-dimensional normed space from ln is at most (2n)^5/6. Finally, using tools from the work of Batson et al in [Twice-Ramanujan sparsifiers. In STOC'09 - Proceedings of the 2009 ACM International Symposium on Theory of Computing, ACM (New York, 2009), 255-262], we give a new proof for a theorem of Kashin and Tzafriri [Some remarks on the restriction of operators to coordinate subspaces. Preprint, 1993] on the norm of restricted matrices.