f-Divergence for convex bodies

TL;DR

This paper introduces f-divergence for convex bodies, establishing its invariance properties and affine isoperimetric inequalities, and connects it to existing concepts like affine surface area.

Contribution

It defines f-divergence for convex bodies, proves its invariance and valuation properties, and links it to affine surface areas within the L_p Brunn Minkowski theory.

Findings

f-divergences are SL(n) invariant valuations

established an affine isoperimetric inequality for f-divergences

generalized affine surface area is a special case of f-divergence

Abstract

We introduce f-divergence, a concept from information theory and statistics, for convex bodies in R^n. We prove that f-divergences are SL(n) invariant valuations and we establish an affine isoperimetric inequality for these quantities. We show that generalized affine surface area and in particular the L_p affine surface area from the L_p Brunn Minkowski theory are special cases of f-divergences.