Small ball probability estimates, psi(2)-behavior and the hyperplane conjecture
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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 2-body has bounded isotropic constant, with a slightly better estimate: If K is a symmetric convex body in Rn such that {norm of matrix} , {norm of matrix}q {norm of matrix} , {norm of matrix}2 for every Sn - 1 and every q 2, then LK C sqrt(log ), where C > 0 is an absolute constant. 2009.