The first variation of the total mass of log-concave functions and related inequalities

On the class of log-concave functions on Rn, endowed with a suitable algebraic structure, we study the rst variation of the total mass functional, which corresponds to the volume of convex bodies when restricted to the subclass of characteristic functions. We prove some integral representation formulae for such rst variation, which lead to dene in a natural way the notion of area measure for a log-concave function. In the same framework, we obtain a functional counterpart of Minkowski rst inequality for convex bodies; as corollaries, we derive a functional form of the isoperimetric inequality, and a family of logarithmic-type Sobolev inequalities with respect to log-concave probability measures. Finally, we propose a suitable functional version of the classical Minkowski problem for convex bodies, and prove some partial results towards its solution.