Restricted Invertibility and the Banach-Mazur distance to the cube

TL;DR

This paper advances the understanding of Banach space geometry by refining invertibility principles, providing new proofs for key theorems, and improving bounds on the Banach-Mazur distance to the cube.

Contribution

It introduces a normalized restricted invertibility principle, offers new proofs for the Dvoretzky-Rogers factorization and Kashin-Tzafriri theorems, and improves bounds on the Banach-Mazur distance to the cube.

Findings

Established a normalized restricted invertibility principle.

Provided a new proof of the proportional Dvoretzky-Rogers factorization theorem.

Improved the estimate for the Banach-Mazur distance to the cube to (2n)^(5/6).

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.