  We show that the fundamental objects of the $L_p$-Brunn-Minkowski theory,
namely the $L_p$-affine surface areas for a convex body, are closely related to
information theory: they are exponentials of R\'enyi divergences of the cone
measures of a convex body and its polar.
  We give geometric interpretations for all R\'enyi divergences $D_\alpha$, not
just for the previously treated special case of relative entropy which is the
case $\alpha =1$. Now, no symmetry assumptions are needed and, if at all, only
very weak regularity assumptions are required.
  Previously, the relative entropies appeared only after performing second
order expansions of certain expressions. Now already first order expansions
makes them appear. Thus, in the new approach we detect "faster" details about
the boundary of a convex body.
