Moment measures

摘要

With any convex function psi on a finite-dimensional linear space X such that psi goes to +infinity at infinity, we associate a Borel measure mu on X*. The measure mu is obtained by pushing forward the measure e(-psi(x)) dx under the differential of psi. 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. (C) 2015 Elsevier Inc. All rights reserved.