2020/06/30 by Liran Rotem, Rotem, Liran · 1 citation
Mathematics · #Analytic and geometric function theory #FOS: Mathematics #Functional Analysis (math.FA) #Geometric Analysis and Curvature Flows #Metric Geometry (math.MG) #Point processes and geometric inequalities
paper · pdf · doi:10.48550/arxiv.2006.16933
openalex publication_date 2020/06/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 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.