Relative entropy of cone measures and $L_p$ centroid bodies

TL;DR

This paper introduces a new affine invariant called mi, derived from cone measures and $L_p$-centroid bodies, leading to new inequalities and insights in convex geometry.

Contribution

The paper defines mi as a new affine invariant and explores its properties, establishing new affine isoperimetric and information inequalities for convex bodies.

Findings

Established mi as a limit of normalized $L_p$-affine surface areas

Proved mi as a relative entropy of cone measures

Derived new affine isoperimetric inequalities and an information inequality

Abstract

Let $K$ be a convex body in $\mathbb R^n$. We introduce a new affine invariant, which we call $\Omega_K$, that can be found in three different ways: as a limit of normalized $L_p$-affine surface areas, as the relative entropy of the cone measure of $K$ and the cone measure of $K^\circ$, as the limit of the volume difference of $K$ and $L_p$-centroid bodies. We investigate properties of $\Omega_K$ and of related new invariant quantities. In particular, we show new affine isoperimetric inequalities and we show a "information inequality" for convex bodies.