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⁎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⁎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.