Topological Stability and Latschev-type Reconstruction Theorems for CAT(κ) Spaces
This work addresses the problem of homotopy type reconstruction and Gromov–Hausdorff (GH) stability for compact CAT(κ) spaces. Classical Euclidean-based sampling methods fail when the reach vanishes; to overcome this, we propose a Vietoris–Rips complex construction grounded in the intrinsic path metric and introduce the novel notion of “restricted distortion”, thereby relaxing stringent sampling assumptions such as positive weak feature size or μ-reachability. Our method achieves, for the first time, stable homotopy reconstruction for spaces with zero reach but finite distortion—e.g., cusps and corners—and establishes an analogue of Latschev’s theorem for CAT(κ) spaces, rigorously proving topological stability and GH finiteness. Experiments validate the approach on canonical nonsmooth Euclidean subsets. The framework provides a geometrically robust discretization theory for nonlinear shape analysis.