On the class of log-concave functions on Rn, endowed with a suitable
algebraic structure, we study the first 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 first variation, which lead to define 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 first 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.Comment: 36 page