### Abstract

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.

Original language | English (US) |
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Pages (from-to) | 201-218 |

Number of pages | 18 |

Journal | Mathematika |

Volume | 60 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1 2014 |

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### Keywords

- 15A60
- 46B20 (primary)
- MSC (2010): 15A09

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Mathematika*,

*60*(1), 201-218. https://doi.org/10.1112/S0025579313000144