Rényi divergence and [formula omitted]-affine surface area for convex bodies

Abstract

We show that the fundamental objects of the Lp-Brunn–Minkowski theory, namely the Lp-affine surface areas for a convex body, are closely related to information theory: they are exponentials of Rényi divergences of the cone measures of a convex body and its polar.We give geometric interpretations for all Rényi divergences Dα, not just for the previously treated special case of relative entropy which is the case α=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.