Functional version of Lp-affine surface area and entropy inequalities, preprint

In contemporary convex geometry, the rapidly developing Lp-Brunn Minkowski theory is a modern analogue of the classical Brunn Minkowski theory. A cornerstone of this theory is the Lp-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 Lp-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. 1 Introduction. The starting point of this paper is a reverse log-Sobolev inequality for log concave func-tions due to Artstein, Klartag, Schütt and Werner [3]. We first recall the usual log-Sobolev inequality. Let γn be the standard Gaussian measure on Rn. The log-Sobolev inequality, due to Gross [18] (see also [14, 30]), asserts that for every probability measur