Moment Measures

TL;DR

This paper introduces a bijective correspondence between a class of convex functions and finite Borel measures on dual spaces, linking concepts from convex analysis, complex geometry, and the Minkowski problem.

Contribution

It defines a new class of convex functions called essentially-continuous functions and establishes a bijection with certain finite Borel measures, connecting multiple mathematical fields.

Findings

Establishes a bijection between convex functions and measures with specific properties.

Links the construction to toric Kähler-Einstein metrics and Prékopa's inequality.

Provides a new perspective on the Minkowski problem in convex 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.