Restricted Invertibility and the Banach-Mazur distance to the cube

@article{Youssef2014RestrictedIA,
  title={Restricted Invertibility and the Banach-Mazur distance to the cube},
  author={Pierre Youssef},
  journal={Mathematika},
  year={2014},
  volume={60},
  pages={201-218}
}
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… 

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