A remark on the slicing problem

The purpose of this article is to describe a reduction of the slicing problem to the study of the parameter I1(K,Zq∘(K))=∫K‖〈⋅,x〉‖Lq(K)dx. We show that an upper bound of the form I1(K,Zq∘(K))⩽C1qsnLK2, with 1/2⩽s⩽11/2⩽s⩽1, leads to the estimateLn⩽C2n4lognq1−s2, where Ln:=max{LK:K is an isotropic convex body in Rn}Ln:=max{LK:K is an isotropic convex body in Rn}.