Moment Measures
Functional Analysis
2013-04-03 v1 Symplectic Geometry
Abstract
With any convex function F on a finite-dimensional linear space X such that F goes to infinity at infinity, we associate a Borel measure on the dual space X*. This measure is obtained by pushing forward the measure exp(-F(x))dx under the differential of F. We propose a class of convex functions - the essentially-continuous, convex functions - for which the above correspondence is in fact a bijection onto the class of finite Borel measures whose barycenter is at the origin and whose support linearly spans the entire space X*. The construction is related to toric Kahler-Einstein metrics in complex geometry, to Pr\'ekopa's inequality, and to the Minkowski problem in convex geometry.
Cite
@article{arxiv.1304.0630,
title = {Moment Measures},
author = {Dario Cordero-Erausquin and Bo'az Klartag},
journal= {arXiv preprint arXiv:1304.0630},
year = {2013}
}