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 Kahler-Einstein metrics in complex geometry, to Prekopa's inequality, and to the Minkowski problem in convex geometry.