Pinsker inequalities and related Monge-Ampère equations for log concave functions

Abstract: In this paper we further develop the theory of f-divergences for log-concave functions and their related inequalities. We establish Pinsker inequalities and new affine invariant entropy inequalities. We obtain new inequalities on functional affine surface area and lower and upper bounds for the Kullback-Leibler divergence in terms of functional affine surface area. The functional inequalities lead to new inequalities for L_p-affine surface areas for convex bodies.