  We introduce a new class of (not necessarily convex) bodies and show, among
other things, that these bodies provide yet another link between convex
geometric analysis and information theory. Namely, they give geometric
interpretations of the relative entropy of the cone measures of a convex body
and its polar and related quantities.
  Such interpretations were first given by Paouris and Werner for symmetric
convex bodies in the context of the $L_p$-centroid bodies. There, the relative
entropies appear after performing second order expansions of certain
expressions. Now, no symmetry assumptions are needed. Moreover, using the new
bodies, already first order expansions make the relative entropies appear.
Thus, these bodies detect "faster" details of the boundary of a convex body
than the $L_p$-centroid bodies.
