On the reverse Lp-Busemann-Petty centroid inequality

The volume of the L-p-centroid body of a convex body K subset of R-d is a convex function of a time-like parameter when each chord of K parallel to a fixed direction moves with constant speed. This fact is used to study extrema of some affine invariant functionals involving the volume of the L-p-centroid body and related to classical open problems like the slicing problem. Some variants of the L-p-Busemann-Petty centroid inequality are established. The reverse form of these inequalities is proved in the two-dimensional case.