{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/gromov-hausdorff-distances-on-p-metric-spaces","title":"On $p$-metric spaces and the $p$-Gromov-Hausdorff distance","arxiv_id":"1912.00564","date":"2019-12-02","proceeding":null,"authors":["Facundo Mémoli","Zhengchao Wan"],"abstract":"For each given $p\\in[1,\\infty]$ we investigate certain sub-family $\\mathcal{M}_p$ of the collection of all compact metric spaces $\\mathcal{M}$ which are characterized by the satisfaction of a strengthened form of the triangle inequality which encompasses, for example, the strong triangle inequality satisfied by ultrametric spaces. We identify a one parameter family of Gromov-Hausdorff like distances $\\{d_{\\mathrm{GH}}^{\\scriptscriptstyle{(p)}}\\}_{p\\in[1,\\infty]}$ on $\\mathcal{M}_p$ and study geometric and topological properties of these distances as well as the stability of certain canonical projections $\\mathfrak{S}_p:\\mathcal{M}\\rightarrow \\mathcal{M}_p$. For the collection $\\mathcal{U}$ of all compact ultrametric spaces, which corresponds to the case $p=\\infty$ of the family $\\mathcal{M}_p$, we explore a one parameter family of interleaving-type distances and reveal their relationship with $\\{d_{\\mathrm{GH}}^{\\scriptscriptstyle{(p)}}\\}_{p\\in[1,\\infty]}$.","url_abs":"https://arxiv.org/abs/1912.00564v4","url_pdf":"https://arxiv.org/pdf/1912.00564v4.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"gromov-hausdorff-distances-on-p-metric-spaces","repo_url":"https://github.com/ndag/ultrametrics","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":null},{"paper_slug":"gromov-hausdorff-distances-on-p-metric-spaces","repo_url":"https://github.com/xuegong-git/pygh","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}