Divergence for s-concave and log concave functions

TL;DR

This paper introduces new entropy inequalities for s-concave and log-concave functions, generalizing existing inequalities and exploring their properties and applications in convex geometry.

Contribution

It establishes novel entropy inequalities for s-concave and log-concave functions, including properties like affine invariance and applications to convex bodies.

Findings

New entropy inequalities for s-concave and log-concave functions

Introduction of f-divergence and relative entropy concepts for these functions

Applications in convex body theory

Abstract

We prove new entropy inequalities for log concave and s-concave functions that strengthen and generalize recently established reverse log Sobolev and Poincare inequalities for such functions. This leads naturally to the concept of f-divergence and, in particular, relative entropy for s-concave and log concave functions. We establish their basic properties, among them the affine invariant valuation property. Applications are given in the theory of convex bodies.