Small ball probability estimates, ψ2behavior and the hyperplane conjecture
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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 ψ2body has bounded isotropic constant, with a slightly better estimate: If K is a symmetric convex body in Rn such that {norm of matrix} 〈 ṡ, θ 〉 {norm of matrix}q ≤ β {norm of matrix} 〈 ṡ, θ 〉 {norm of matrix}2 for every θ ∈ Sn  1 and every q ≥ 2, then LK ≤ C β sqrt(log β), where C > 0 is an absolute constant. © 2009.
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Dafnis, N., & Paouris, G.
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Hyperplane Conjecture

Psi(2)bodies

Small Ball Probability
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