  We prove a normalized version of the restricted invertibility principle
obtained by Spielman-Srivastava. Applying this result, we get a new proof of
the proportional Dvoretzky-Rogers factorization theorem recovering the best
current estimate. As a consequence, we also recover the best known estimate for
the Banach-Mazur distance to the cube: the distance of every n-dimensional
normed space from \ell_{\infty}^n is at most (2n)^(5/6). Finally, using tools
from the work of Batson-Spielman-Srivastava, we give a new proof for a theorem
of Kashin-Tzafriri on the norm of restricted matrices.
