Small ball probability estimates, 2-behavior and the hyperplane conjecture

摘要

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 2-body has bounded isotropic constant, with a slightly better estimate: If K is a symmetric convex body in Rn such that q2 for every Sn1 and every q 2, then LKog, whereC >0 is an absolute constant.