Centroid Bodies and the Logarithmic Laplace Transform - A Unified Approach

Abstract

We unify and slightly improve several bounds on the isotropic constant of high-dimensional convex bodies; in particular, a linear dependence on the body's psi-2 constant is obtained. Along the way, we present some new bounds on the volume of L_p-centroid bodies and yet another equivalent formulation of Bourgain's hyperplane conjecture. Our method is a combination of the L_p-centroid body technique of Paouris and the logarithmic Laplace transform technique of the first named author.