Relative entropy of cone measures and L p centroid bodies

Let K be a convex body in ℝn. We introduce a new affine invariant, which we call ΩK, that can be found in three different ways: as a limit of normalized Lp-affine surface areas; as the relative entropy of the cone measure of K and the cone measure of K°; as the limit of the volume difference of K and Lp-centroid bodies. We investigate properties of ΩK and of related new invariant quantities. In particular, we show new affine isoperimetric inequalities and we show an ‘information inequality’ for convex bodies.