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Modifications of the Continuous Gromov--Hausdorff Distance

  • A. A. Vikhrov
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Abstract

This work studies modifications of the Gromov--Hausdorff distance in which the infimum is taken over pairs of morphisms of a given class FF --- the class of morphisms of a subcategory of the category of metric spaces (the FF-Gromov--Hausdorff distance). We prove that the FF-distance depends Lipschitz-continuously on the class FF in the Hausdorff metric on morphisms. As a consequence, for second countable Ck-manifolds equipped with a metric compatible with the topology, the smooth distance coincides with the continuous one: dGHCk = dGHC. We also introduce the partially continuous and partially locally constant distances (continuity is weakened to continuity on an open subset of the support of full measure) and prove that on the class of metric spaces they coincide with the classical Gromov--Hausdorff distance.

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Cite

BibTeXRIS
BibTeX
@misc{vikhrov2026modifications,
  title = {{Modifications of the Continuous Gromov--Hausdorff Distance}},
  author = {A. A. Vikhrov},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.22201},
  eprint = {2609.22201},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.22201},
}
RIS
TY  - UNPB
TI  - Modifications of the Continuous Gromov--Hausdorff Distance
AU  - A. A. Vikhrov
PY  - 2026
DA  - 2026/09/01
PB  - arXiv
DO  - 10.48550/arxiv.2609.22201
UR  - https://arxiv.org/abs/2609.22201
AB  - This work studies modifications of the Gromov--Hausdorff distance in which the infimum is taken over pairs of morphisms of a given class FF --- the class of morphisms of a subcategory of the category of metric spaces (the FF-Gromov--Hausdorff distance). We prove that the FF-distance depends Lipschitz-continuously on the class FF in the Hausdorff metric on morphisms. As a consequence, for second countable C^k-manifolds equipped with a metric compatible with the topology, the smooth distance coincides with the continuous one: dGHC_k = dGHC. We also introduce the partially continuous and partially locally constant distances (continuity is weakened to continuity on an open subset of the support of full measure) and prove that on the class of metric spaces they coincide with the classical Gromov--Hausdorff distance.
ER  -