Surface area measures of log-concave functions

TL;DR

This paper explores the surface area measure of log-concave functions, providing a first variation formula and extending the concept to an L^p setting, including a Minkowski existence theorem for even measures.

Contribution

It introduces a unified framework for surface area measures of log-concave functions, generalizes to L^p spaces, and proves a Minkowski existence theorem in this broader context.

Findings

Proved a first variation formula for log-concave functions.

Extended surface area measure to the L^p setting.

Established a Minkowski existence theorem for even measures in the L^p context.

Abstract

This paper's origins are in two papers: One by Colesanti and Fragal\`a 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 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 L^p-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.