math.MG — Metric Geometry
FINITE HAUSDORFF HYPERSPACES AND GROMOV–HAUSDORFF GEOMETRY
For a nonempty finite metric space X, let Exp(X) be the space of its nonempty subsets equipped with the Hausdorff metric. We prove that the isometry type of Exp(X) determines X when X is strongly rigid, meaning that distinct unordered pairs of distinct points have dis- tinct distances. We also prove that the hyperspace operation preserves the ordinary Gromov–Hausdorff distance locally under explicit separa- tion conditions on the distance spectra. For finite ultrametric spaces, the hyperspace operation is an isometric embedding with respect to the Gromov–Hausdorff ultrametric. In contrast, we construct finite ultra- metric spaces for which the ordinary Gromov–Hausdorff distance strictly decreases under the hyperspace operation. Examples of equal cardinality exist in every cardinality at least six and realize every ratio in [1/2, 1). Their distances remain constant after the first hyperspace iteration. An explicit perturbation also yields strict contraction for finite strongly rigid metric spaces with all triangle inequalities strict.