Surface area measures of log-concave functions

This paper’s origins are in two papers: One by Colesanti and Fragalà studying the surface area measure of a log-concave function, one Cordero-Erausquin Klartag regarding moment convex function. These notions same, this paper we continue same construction as well its generalization. In first half prove variation formula for integral functions under minimal optimal conditions. We also explain why result is common generalization known theorems from above papers. second extend definition functional to Lp-setting, generalizing classic Lutwak. generalized setting Minkowski existence theorem even measures. partial extension that handled case p = 1 not necessarily