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Gromov-Hausdorff class: its completeness and cloud geometry

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arxiv 2110.06101 v1 pith:V55SBCBQ submitted 2021-10-12 math.MG

Semeon A. Bogaty , Alexey A. Tuzhilin This is my paper · ORCID
classification math.MG
keywords spaceclasscloudcloudsgromov-hausdorffcompletenessconsistingdistance
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The paper is devoted to the study of the Gromov-Hausdorff proper class, consisting of all metric spaces considered up to isometry. In this class, a generalized Gromov-Hausdorff pseudometric is introduced and the geometry of the resulting space is investigated. The first main result is a proof of the completeness of the space, i.e., that all fundamental sequences converge in it. Then we partition the space into maximal proper subclasses consisting of spaces at a finite distance from each other. We call such subclasses clouds. A multiplicative similarity group operates on clouds, multiplying all the distances of each metric space by some positive number. We present examples of similarity mappings transferring some clouds into another ones. We also show that if a cloud contains a space that remains at zero distance from itself under action of all similarities, then such a cloud contracted to this space. In the final part, we investigate subsets of the real line with respect to their behavior under various similarities.

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Cited by 6 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. New geodesic lines in the Gromov-Hausdorff class lying in the cloud of the real line

    math.MG 2025-04 conditional novelty 7.0 of 10

    For unbounded A subset of R and bounded X, the curve A times_{l1} (tX) is a geodesic ray, and d_GH(Z^n, lambda Z^n) is at least 1/2 for every lambda > 1.

  2. Gromov-Hausdorff Geometry of Metric Trees

    math.MG 2024-12 conditional novelty 7.0 of 10

    For subsets of the real line and of infinite metric trees satisfying a boundary condition, the Gromov-Hausdorff distance to the ambient space equals the Hausdorff distance, and the canonical Hausdorff geodesic is a sh...

  3. On the Gromov-Hausdorff distance between the cloud of bounded metric spaces and a cloud with nontrivial stabilizer

    math.MG 2025-05 conditional novelty 6.0 of 10

    The Gromov-Hausdorff distance between the cloud of bounded metric spaces and the cloud containing the real line is infinite, and a criterion for such infinite cloud distances is proved.

  4. Calculating Gromov-Hausdorff distance by means of asymptotic dimension

    math.MG 2025-05 conditional novelty 6.0 of 10

    The paper proves d_GH(Z^2,R^2)=sqrt(2)/2 and d_GH(A,R^2)=1/2 for a comb-shaped set A, using a new theorem that bounds GH distance below via r-disjoint covers and asymptotic dimension.

  5. When the Gromov-Hausdorff distance between finite-dimensional space and its subset is finite?

    math.MG 2024-11 conditional novelty 6.0 of 10

    A subset A of finite-dimensional Euclidean space has finite Gromov-Hausdorff distance to the whole space if and only if A is an epsilon-net for some epsilon.

  6. Ultrametric spaces and clouds

    math.MG 2025-01 conditional novelty 5.0 of 10

    The ultrametrization map U is 1-Lipschitz on all metric spaces, preserves products with dotted connected spaces, and forces mutual exclusion of ultrametric and dotted connected spaces in unbounded clouds.

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