, Volume 16, Issue 6, pp 1274-1290
Date: 06 Dec 2006

On convex perturbations with a bounded isotropic constant

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Abstract.

Let \( K \subset {\user2{\mathbb{R}}}^{n} \) be a convex body and ɛ  > 0. We prove the existence of another convex body \( K' \subset {\user2{\mathbb{R}}}^{n} \) , whose Banach–Mazur distance from K is bounded by 1 + ɛ, such that the isotropic constant of K’ is smaller than \( c \mathord{\left/ {\vphantom {c {{\sqrt \varepsilon }}}} \right. \kern-0em} {{\sqrt \varepsilon }} \) , where c  > 0 is a universal constant. As an application of our result, we present a slight improvement on the best general upper bound for the isotropic constant, due to Bourgain.

The author is a Clay Research Fellow, and was also supported by NSF grant #DMS-0456590.
Received: November 2005; Accepted: February 2006