This is an announcement for the paper "Restricted Invertibility and the Banach-Mazur distance to the cube" by Pierre Youssef.

Abstract: 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.

Archive classification: math.FA

Submitted from: pierre.youssef@univ-mlv.fr

The paper may be downloaded from the archive by web browser from URL

http://front.math.ucdavis.edu/1206.0654

or

http://arXiv.org/abs/1206.0654