Small ball probability estimates, [formula omitted]-behavior and the hyperplane conjecture

Abstract

We 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 ψ 2 -body has bounded isotropic constant, with a slightly better estimate: If K is a symmetric convex body in R n such that ‖ 〈 ⋅ , θ 〉 ‖ q ⩽ β ‖ 〈 ⋅ , θ 〉 ‖ 2 for every θ ∈ S n − 1 and every q ⩾ 2 , then L K ⩽ C β log β , where C > 0 is an absolute constant.