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.