A note on the reduction of the slicing problem to centrally symmetric convex bodies

Abstract

In this paper, we obtain the best possible value of the absolute constant C such that for every isotropic convex body K⊆Rn the following inequality (which was proved by Klartag and reduces the hyperplane conjecture to centrally symmetric convex bodies) is satisfied:LK≤CLKn+2(gK). Here LK denotes the isotropic constant of K, gK its covariogram function, which is log-concave, and, for any log-concave function g, Kn+2(g) is a convex body associated to the log-concave function g, which belongs to a uniparametric family introduced by Ball. In order to obtain this inequality, sharp inclusion results between the convex bodies in this family are obtained whenever g satisfies a better type of concavity than the log-concavity, as gK is, indeed 1n-concave.