Functional versions of L_p-affine surface area and entropy inequalities

In contemporary convex geometry, the rapidly developing L_p-Brunn Minkowski theory is a modern analogue of the classical Brunn Minkowski theory. A cornerstone of this theory is the L_p-affine surface area for convex bodies. Here, we introduce a functional form of this concept, for log concave and s-concave functions. We show that the new functional form is a generalization of the original L_p-affine surface area. We prove duality relations and affine isoperimetric inequalities for log concave and s-concave functions. This leads to a new inverse log-Sobolev inequality for s-concave densities.