Wasserstein barycenters over Riemannian manifolds
Let \(M\) be a compact Riemannian manifold with the space of Borel probability measures on \(M\) denoted by \(P(M)\). The Wasserstein space associated to \(M\) is the metric space \(P(M)\) endowed with the Wasserstein metric. For any probability measure \(\Omega\) on that Wasserstein space, its Wasserstein barycenter is an element of \(P(M)\). The main results of the paper under review give sufficient conditions for the barycenter of \(\Omega\) to be absolutely continuous with respect to the volume measure of~\(M\). These conditions are more general than the ones from earlier work and as applications, the authors are able to obtain versions of the Jensen inequality and a generalized Brunn-Minkowski inequality.
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