Relative entropy of cone measures and $L_p$ centroid bodies

Let $K$ be a convex body in ${\mathbb{R}}^n$. We introduce a new affine invariant, which we call ${\mathrm{\Omega }}_K$, that can be found in three different ways: as a limit of normalized $L_p$-affine surface areas, as the relative entropy of the cone measure of $K$ and the cone measure of $K^{\circ }$, as the limit of the volume difference of $K$ and $L_p$-centroid bodies. We investigate properties of ${\mathrm{\Omega }}_K$ and of related new invariant quantities. In particular, we show new affine isoperimetric inequalities and we show a “information inequality” for convex bodies.