On convex perturbations with a bounded isotropic constant

Let be a convex body and ɛ > 0. We prove the existence of another convex body , whose Banach–Mazur distance from K is bounded by 1 + ɛ, such that the isotropic constant of K’ is smaller than , 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