{"abstract":[{"lang":"eng","text":"We discuss the embeddability of subspaces of the Gromov–Hausdorff space, which consists of isometry classes of compact metric spaces endowed with the Gromov–Hausdorff distance, into Hilbert spaces. These embeddings are particularly valuable for applications to topological data analysis. We prove that its subspace consisting of metric spaces with at most n points has asymptotic dimension n(n−1)∕2. Thus, there exists a coarse embedding of that space into a Hilbert space. On the contrary, if the number of points is not bounded, then the subspace cannot be coarsely embedded into any uniformly convex Banach space and so, in particular, into any Hilbert space. Furthermore, we prove that, even if we restrict to finite metric spaces whose diameter is bounded by some constant, the subspace still cannot be bi-Lipschitz embedded into any finite-dimensional Hilbert space. We obtain both nonembeddability results by finding obstructions to coarse and bi-Lipschitz embeddings in families of isometry classes of finite subsets of the real line endowed with the Euclidean–Hausdorff distance."}],"author":[{"last_name":"Zava","orcid":"0000-0001-8686-1888","id":"c8b3499c-7a77-11eb-b046-aa368cbbf2ad","full_name":"Zava, Nicolò","first_name":"Nicolò"}],"_id":"20867","issue":"8","acknowledgement":"The author was supported by the FWF Grant, Project number I4245-N35. The author would like to thank Thomas Weighill for the helpful discussions around Theorem 3.10, and Takamitsu Yamauchi for bringing to my attention the fundamental reference [35]. Furthermore, the author\r\nis thankful for the detailed and helpful comments of the reviewer of this manuscript.","title":"Coarse and bi-Lipschitz embeddability of subspaces of the Gromov–Hausdorff space into Hilbert spaces","publication_identifier":{"eissn":["1472-2739"],"issn":["1472-2747"]},"publication":"Algebraic & Geometric Topology","file":[{"access_level":"open_access","success":1,"file_name":"2025_AlgebraicGeomTopology_Zava.pdf","creator":"dernst","relation":"main_file","date_updated":"2026-01-05T12:16:38Z","file_id":"20943","date_created":"2026-01-05T12:16:38Z","content_type":"application/pdf","file_size":574389,"checksum":"1e05b4f17a44500ae1ae1e21bc636f6a"}],"year":"2025","type":"journal_article","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","OA_place":"publisher","volume":25,"tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png","short":"CC BY (4.0)"},"publisher":"Mathematical Sciences Publishers","oa":1,"article_processing_charge":"No","citation":{"ista":"Zava N. 2025. Coarse and bi-Lipschitz embeddability of subspaces of the Gromov–Hausdorff space into Hilbert spaces. Algebraic &#38; Geometric Topology. 25(8), 5153–5174.","chicago":"Zava, Nicolò. “Coarse and Bi-Lipschitz Embeddability of Subspaces of the Gromov–Hausdorff Space into Hilbert Spaces.” <i>Algebraic &#38; Geometric Topology</i>. Mathematical Sciences Publishers, 2025. <a href=\"https://doi.org/10.2140/agt.2025.25.5153\">https://doi.org/10.2140/agt.2025.25.5153</a>.","ama":"Zava N. Coarse and bi-Lipschitz embeddability of subspaces of the Gromov–Hausdorff space into Hilbert spaces. <i>Algebraic &#38; Geometric Topology</i>. 2025;25(8):5153-5174. doi:<a href=\"https://doi.org/10.2140/agt.2025.25.5153\">10.2140/agt.2025.25.5153</a>","ieee":"N. Zava, “Coarse and bi-Lipschitz embeddability of subspaces of the Gromov–Hausdorff space into Hilbert spaces,” <i>Algebraic &#38; Geometric Topology</i>, vol. 25, no. 8. Mathematical Sciences Publishers, pp. 5153–5174, 2025.","short":"N. Zava, Algebraic &#38; Geometric Topology 25 (2025) 5153–5174.","apa":"Zava, N. (2025). Coarse and bi-Lipschitz embeddability of subspaces of the Gromov–Hausdorff space into Hilbert spaces. <i>Algebraic &#38; Geometric Topology</i>. Mathematical Sciences Publishers. <a href=\"https://doi.org/10.2140/agt.2025.25.5153\">https://doi.org/10.2140/agt.2025.25.5153</a>","mla":"Zava, Nicolò. “Coarse and Bi-Lipschitz Embeddability of Subspaces of the Gromov–Hausdorff Space into Hilbert Spaces.” <i>Algebraic &#38; Geometric Topology</i>, vol. 25, no. 8, Mathematical Sciences Publishers, 2025, pp. 5153–74, doi:<a href=\"https://doi.org/10.2140/agt.2025.25.5153\">10.2140/agt.2025.25.5153</a>."},"ddc":["500"],"page":"5153-5174","OA_type":"diamond","article_type":"original","project":[{"grant_number":"I04245","name":"Algebraic Footprints of Geometric Features in Homology","call_identifier":"FWF","_id":"26AD5D90-B435-11E9-9278-68D0E5697425"}],"day":"20","language":[{"iso":"eng"}],"department":[{"_id":"HeEd"}],"date_created":"2025-12-29T12:09:09Z","file_date_updated":"2026-01-05T12:16:38Z","arxiv":1,"publication_status":"published","status":"public","date_updated":"2026-01-05T12:19:09Z","date_published":"2025-11-20T00:00:00Z","external_id":{"arxiv":["2303.04730"]},"month":"11","oa_version":"Published Version","doi":"10.2140/agt.2025.25.5153","PlanS_conform":"1","quality_controlled":"1","corr_author":"1","scopus_import":"1","intvolume":"        25","has_accepted_license":"1"}