Small ball probability estimates, ψ2-behavior and the hyperplane conjecture

e introduce a method which leads to upper bounds for the isotropic constant. We prove that a positive answer to the hyperplane conjecture is equivalent to some very strong small probability estimates for the Euclidean norm on isotropic convex bodies. As a consequence of our method, we obtain an alternative proof of the result of J. Bourgain that every ψ2-body has bounded isotropic constant, with a slightly better estimate: If K is a symmetric convex body in Rn such that ·, θ q β ·, θ 2 for every θ ∈ Sn−1 and every q 2, then LK Cβ√logβ, whereC >0 is an absolute constant. © 2009 Published by Elsevier Inc.