Moment measures

D. Cordero-Erausquin, B. Klartag

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish
Pages (from-to)3834-3866
Number of pages33
JournalJournal of Functional Analysis
Issue number12
StatePublished - 15 Jun 2015


  • Moment measure
  • Prékopa theorem
  • Toric Kähler-Einstein metrics


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