Surface area measures of log-concave functions

Abstract

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