Coarse infinite-dimensionality of hyperspaces of finite subsets

Abstract

We consider infinite-dimensional properties in coarse geometry for hyperspaces consisting of finite subsets of metric spaces with the Hausdorff metric. We see that several infinite-dimensional properties are preserved by taking the hyperspace of subsets with at most n points. On the other hand, we prove that, if a metric space contains a sequence of long intervals coarsely, then its hyperspace of finite subsets is not coarsely embeddable into any uniformly convex Banach space. As a corollary, the hyperspace of finite subsets of the real line is not coarsely embeddable into any uniformly convex Banach space. It is also shown that every (not necessarily bounded geometry) metric space with straight finite decomposition complexity has metric sparsification property.

The Gromov-Hausdorff space is not coarsely embeddable into any Hilbert space

“…To prove Theorem 3.1, we intend to apply Theorem 2.7. Following the approach used in [WYZ22], let us first isometrically send an arbitrary finite metric space into a more manageable space. Lemma 3.3 (Kuratowski embedding).…”

“…Thus, D(1, 1) = 12 (according to the notation that max ∅ = −∞), and 2). This construction should be compared with that provided in [WYZ22]. The crucial difference is the last point D(m, n).…”

“…This strategy has been successfully applied to several spaces. For example, in [WYZ22], the space of all finite subsets of a geodesic unbounded metric space equipped with the Hausdorff distance is considered. In topological data analysis, we mention spaces of persistence diagrams with various metrics ([MV21, BW20, Wag21]).…”