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Moment measures. (English) Zbl 1315.32004

Summary: With any convex function \(\psi\) on a finite-dimensional linear space \(X\) such that \(\psi\) goes to \(+\infty\) at infinity, we associate a Borel measure \(\mu\) on \(X^\ast\). The measure \(\mu\) is obtained by pushing forward the measure \(e^{- \psi(x)} d x\) 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 \(X^\ast\). The construction is related to toric Kähler-Einstein metrics in complex geometry, to Prékopa’s inequality, and to the Minkowski problem in convex geometry.

MSC:

32F17 Other notions of convexity in relation to several complex variables
32Q20 Kähler-Einstein manifolds
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