---
OA_place: publisher
OA_type: diamond
PlanS_conform: '1'
_id: '20867'
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.
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."
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Nicolò
  full_name: Zava, Nicolò
  id: c8b3499c-7a77-11eb-b046-aa368cbbf2ad
  last_name: Zava
  orcid: 0000-0001-8686-1888
citation:
  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>
  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>
  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>.
  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.
  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.
  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>.
  short: N. Zava, Algebraic &#38; Geometric Topology 25 (2025) 5153–5174.
corr_author: '1'
date_created: 2025-12-29T12:09:09Z
date_published: 2025-11-20T00:00:00Z
date_updated: 2026-01-05T12:19:09Z
day: '20'
ddc:
- '500'
department:
- _id: HeEd
doi: 10.2140/agt.2025.25.5153
external_id:
  arxiv:
  - '2303.04730'
file:
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  creator: dernst
  date_created: 2026-01-05T12:16:38Z
  date_updated: 2026-01-05T12:16:38Z
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has_accepted_license: '1'
intvolume: '        25'
issue: '8'
language:
- iso: eng
month: '11'
oa: 1
oa_version: Published Version
page: 5153-5174
project:
- _id: 26AD5D90-B435-11E9-9278-68D0E5697425
  call_identifier: FWF
  grant_number: I04245
  name: Algebraic Footprints of Geometric Features in Homology
publication: Algebraic & Geometric Topology
publication_identifier:
  eissn:
  - 1472-2739
  issn:
  - 1472-2747
publication_status: published
publisher: Mathematical Sciences Publishers
quality_controlled: '1'
scopus_import: '1'
status: public
title: Coarse and bi-Lipschitz embeddability of subspaces of the Gromov–Hausdorff
  space into Hilbert spaces
tmp:
  image: /images/cc_by.png
  legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
  name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
  short: CC BY (4.0)
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 25
year: '2025'
...