This is an announcement for the paper "The area measure of log-concave functions and related inequalities" by Andrea Colesanti and Ilaria Fragala.
Abstract: On the class of log-concave functions on $\R^n$, 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.
Archive classification: math.FA math.MG
Remarks: 36 pages
Submitted from: colesant@math.unifi.it
The paper may be downloaded from the archive by web browser from URL
http://front.math.ucdavis.edu/1112.2555
or