  In this paper we prove different functional inequalities extending the
classical Rogers-Shephard inequalities for convex bodies. The original
inequalities provide an optimal relation between the volume of a convex body
and the volume of several symmetrizations of the body, such as, its difference
body. We characterize the equality cases in all these inequalities. Our method
is based on the extension of the notion of a convolution body of two convex
sets to any pair of log-concave functions and the study of some geometrical
properties of these new sets.
