Let K be a convex body in ℝ n. We introduce a new affine invariant, which we call Ω 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°;as the limit of the volume difference of K ̊ and L p-centroid bodies. We investigate properties of Ω K and of related new invariant quantities. In particular, we show new affine isoperimetric inequalities and we show an 'information inequality' for convex bodies.
Bibliographical noteFunding Information:
The first author was partially supported by an NSF grant. The second author was partially supported by an NSF grant, an FRG-NSF grant and a BSF grant.