Optimal partial transport for metric pairs

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, $\mathcal{M}_p(X,A)$, are complete, separable and geodesic whenever the underlying space, $X$, is so. We also prove that, for $p>1$, $\mathcal{M}_p(X,A)$ preserves the property of being non-branching, and for $p=2$ it preserves non-negative curvature in the Alexandrov sense. Finally, we prove isometric embeddings of generalised spaces of persistence diagrams $\mathcal{D}_p(X,A)$ into the corresponding spaces $\mathcal{M}_p(X,A)$, 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 $\mathcal{M}_p(X,A)$, including the fact that $\mathcal{D}_2(X,A)$ is an Alexandrov space of non-negative curvature whenever $X$ is a proper non-negatively curved Alexandrov space.