  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.
