Moment measures

With any convex function ψ on a finite-dimensional linear space X such that ψ goes to +∞ at infinity, we associate a Borel measure μ on X^*. The measure μ is obtained by pushing forward the measure e^-ψ(x)dx under the differential of ψ. 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 spans X^*. The construction is related to toric Kähler-Einstein metrics in complex geometry, to Prékopa's inequality, and to the Minkowski problem in convex geometry.