Divergence for s-concave and log concave functions

We prove new entropy inequalities for log concave and s-concave functions that strengthen and generalize recently established reverse log Sobolev and Poincaré 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.