Optimal Partial Transport for Metric Pairs

Abstract In this article we study Figalli and Gigli’s formulation of optimal transport between non-negative Radon measures in the setting of metric pairs. We carry over classical characterisations of optimal plans to this setting and prove that the resulting spaces of measures, $$\mathcal {M}_p(X,A)$$ M p ( X , A ) , are complete, separable and geodesic whenever the underlying space, X , is so. We also prove that, for $$p>1$$ p > 1 , $$\mathcal {M}_p(X,A)$$ M p ( X , A ) preserves the property of being non-branching, and for $$p=2$$ p = 2 it preserves non-negative curvature in the Alexandrov sense. Finally, we prove isometric embeddings of generalised spaces of persistence diagrams $$\mathcal {D}_p(X,A)$$ D p ( X , A ) into the corresponding spaces $$\mathcal {M}_p(X,A)$$ M p ( X , A ) , generalising a result by Divol and Lacombe. As an application of this framework, we show that several known geometric properties of spaces of persistence diagrams follow from those of $$\mathcal {M}_p(X,A)$$ M p ( X , A ) , including the fact that $$\mathcal {D}_2(X,A)$$ D 2 ( X , A ) is an Alexandrov space of non-negative curvature whenever X is a proper non-negatively curved Alexandrov space.