Relative entropies for convex bodies

TL;DR

This paper introduces a new class of bodies linking convex geometric analysis and information theory, providing geometric interpretations of relative entropy without symmetry assumptions, and detecting boundary details more rapidly.

Contribution

It presents a novel class of bodies that generalize previous concepts, enabling direct geometric interpretations of relative entropy for convex bodies and their polars.

Findings

Provides geometric interpretations of relative entropy for convex bodies.

Eliminates symmetry assumptions required in previous interpretations.

Detects boundary details of convex bodies more rapidly than existing methods.

Abstract

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.