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FINITE HAUSDORFF HYPERSPACES AND GROMOV–HAUSDORFF GEOMETRY

Primarily AI-generated textHuman understanding: some partsmath.MG — Metric Geometrymath.GN — General Topology

Contributed by Yoshito Ishiki ↗

SubmitterYoshito Ishiki

Version 1 / Sep 30, 2026 / CC BY 4.0

Abstract

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.

Provenance statement

This work was developed with assistance from OpenAI Codex for mathematical exploration, the search for proof constructions and counterexamples, proof checking, finite computations, language editing, and LaTeX preparation. The author proposed using the closed-ball profile $\beta_A^X(r)$ to recognize singleton subsets in the Hausdorff hyperspace and directed the investigation. The author selected the material included in the manuscript and takes responsibility for its content. The present manuscript revises the author's earlier AI-Assisted Research Report: Finite Hausdorff Hyperspaces and Gromov–Hausdorff Geometry, version 1.0, published on Zenodo on 13 August 2026 (DOI: 10.5281/zenodo.21913552), which is cited in the manuscript.

References

  1. Burago, Dmitri; Burago, Yuri; Ivanov, Sergei. A Course in Metric Geometry. American Mathematical Society, vol. 33DOI
  2. Memoli, Facundo; Smith, Zane; Wan, Zhengchao. The Gromov–Hausdorff Distance between Ultrametric Spaces: Its Structure and Computation
  3. Mikhailov, Ivan A. Hausdorff Mapping: 1-Lipschitz and Isometry Properties. vol. 73, no. 6, pp. 211–216DOI
  4. Zarichnyi, Ihor. Gromov–Hausdorff Ultrametric

Version history

  1. v1Initial depositCurrentSep 30, 2026