Renyi divergence and Lp-affine surface area for convex bodies

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 Renyi divergences of the cone measures of a convex body and its polar. We give geometric interpretations for all Renyi 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 make them appear. Thus, in the new approach we detect faster details about the boundary of a convex body. (C) 2012 Elsevier Inc. All rights reserved.