Centroid bodies and the logarithmic Laplace transform – A unified approach

We unify and slightly improve several bounds on the isotropic constant of high-dimensional convex bodies; in particular, a linear dependence on the bodyʼs ψ2 constant is obtained. Along the way, we present some new bounds on the volume of Lp-centroid bodies and yet another equivalent formulation of Bourgainʼs hyperplane conjecture. Our method is a combination of the Lp-centroid body technique of Paouris and the logarithmic Laplace transform technique of the first named author.