Centroid Bodies and the Logarithmic Laplace Transform - A Unified Approach

TL;DR

This paper unifies and improves bounds on the isotropic constant of high-dimensional convex bodies using a novel combination of centroid body and Laplace transform techniques, offering new insights into volume bounds and hyperplane conjecture reformulations.

Contribution

It introduces a unified approach combining centroid bodies and Laplace transforms, improving bounds and providing new formulations related to the hyperplane conjecture.

Findings

Linear dependence on the psi-2 constant for isotropic bounds

New bounds on volume of L_p-centroid bodies

An equivalent formulation of Bourgain's hyperplane conjecture

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.