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Lectures on Hausdorff and Gromov-Hausdorff Distance Geometry

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arxiv 2012.00756 v1 pith:LVADQWFB submitted 2020-12-01 math.MG

Alexey A. Tuzhilin This is my paper · ORCID
classification math.MG
keywords gromov-hausdorffdistancespacescorrespondencesnumbertermsgh-spacemetric
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The course was given at Peking University, Fall 2019. We discuss the following subjects: (1) Introduction to general topology, hyperspaces, metric and pseudometric spaces, graph theory. (2) Graphs in metric spaces, minimum spanning tree, Steiner minimal tree, Gromov minimal filling. (3) Hausdorff distance, Vietoris topology, Limits theory, inheritance of completeness, total boundedness, compactness by hyperspaces. (4) Gromov-Hausdorff distance, triangle inequality, positive definiteness for isometry classes of compact spaces, counterexample for boundedly compact spaces. (5) Gromov-Hausdorff distance for separable spaces in terms of their isometric images in \ell_\infty, correspondences, Gromov-Hausdorff distance in terms of correspondences. (6) Epsilon-isometries and Gromov-Hausdorff distance. (7) Irreducible correspondences and Gromov-Hausdorff distance. (8) Gromov-Hausdorff convergence, inheritance of metric and topological properties while Gromov-Hausdorff convergence. (9) Gromov-Hausdorff space (GH-space), optimal correspondences, existence of closed optimal correspondences for compact metric spaces, GH-space is geodesic. (10) Cover number, packing number, total boundedness, completeness, and separability of GH-space. (11) mst-spectrum in terms of GH-distances to simplexes, Steiner problem in GH-space. (12) GH-distance to simplexes with more points, GH-distance to simplexes with at most the same number of points. (13) Generalized Borsuk problem, solution of Generalized Borsuk problem in terms of GH-distances, clique covering number and chromatic number of simple graphs, their dualities, calculating these numbers in terms of GH-distances.

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Cited by 4 Pith papers

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

  1. On the Existence of Minimal Parametric Networks

    math.MG 2026-07 accept novelty 6.0 of 10

    Compactly equipped pseudometric spaces—where each closed ball carries a compact topology making the distance lower semicontinuous—always contain minimal parametric networks of any type.

  2. Quasi-isometric modification of Gromov-Hausdorff distance

    math.MG 2026-02 conditional novelty 6.0 of 10

    A quasi-isometric analogue of the Gromov–Hausdorff distance is defined, shown to sit between GH and pointed-GH convergence, metrizable, and path-connected on quasi-isometry classes.

  3. Gromov-Hausdorff distance between chromatic metric pairs and stability of the six-pack

    math.MG 2025-07 conditional novelty 6.0 of 10

    Introduces the C-constrained Gromov-Hausdorff distance for chromatic metric pairs and proves that all six diagrams in a six-pack are stable in bottleneck distance with respect to it, up to a factor of 2.

  4. 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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