The purpose of this article is to describe a reduction of the slicing problem to the study of the parameter I_1(K,Z_q^o(K))=\int_K ||< :, x> ||_{ L_q(K)}dx. We show that an upper bound of the form I_1(K,Z_q^o(K))\leq C_1q^s\sqrt{ n}L_K^2, with 1/2\leq s\leq 1, leads to the estimate L_n\leq \frac{ C_2\sqrt[4]{ n}log(n)} { q^{ (1-s)/2}}, where L_n:= max { L_K : K is an isotropic convex body in R^n}.
Categories: math.FA math.MG