On convex perturbations with a bounded isotropic constant

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 $c \mathord{\left/ {\vphantom {c {{\sqrt \varepsilon }}}} \right. \kern-\nulldelimiterspace} {{\sqrt \varepsilon }}$, 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.