Small ball probability estimates, ψ2-behavior and the hyperplane conjecture Academic Article uri icon

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 ψ2-body 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.

author list (cited authors)

  • Dafnis, N., & Paouris, G.

citation count

  • 24

publication date

  • March 2010