Geometry of generalised spaces of persistence diagrams and optimal partial transport for metric pairs

In this thesis, we study the geometry of two families of metric spaces that can be defined over a metric pair. We first focus on generalised spaces of persistence diagrams over metric pairs. We prove that the construction of these metric spaces is functorial and preserves certain geometric properties of the underlying space, namely completeness, separability, geodesicity, and non-negative curvature in the sense of Alexandrov. We also study the continuity of these constructions with respect to Gromov--Hausdorff convergence. We then move on to spaces of Radon measures endowed with the optimal partial transport metrics introduced by Figalli and Gigli. We adapt results from Figalli and Gigli’s work to the class of proper metric pairs. Furthermore, we prove that when endowed with the -optimal partial transport distance, the resulting space of Radon measures is a non-negatively curved Alexandrov space, whenever the underlying space has the same property. This result is new, even in the Euclidean setting considered by Figalli and Gigli. Finally, in an appendix, we study basic properties of Gromov–Hausdorff convergence for metric pairs. We prove that this convergence is metrisable in the context of proper metric pairs, and present versions of the classical embedding, completeness, and precompactness theorems.