---
_id: '12763'
abstract:
- lang: eng
  text: 'Kleinjohann (Archiv der Mathematik 35(1):574–582, 1980; Mathematische Zeitschrift
    176(3), 327–344, 1981) and Bangert (Archiv der Mathematik 38(1):54–57, 1982) extended
    the reach rch(S) from subsets S of Euclidean space to the reach rchM(S) of subsets
    S of Riemannian manifolds M, where M is smooth (we’ll assume at least C3). Bangert
    showed that sets of positive reach in Euclidean space and Riemannian manifolds
    are very similar. In this paper we introduce a slight variant of Kleinjohann’s
    and Bangert’s extension and quantify the similarity between sets of positive reach
    in Euclidean space and Riemannian manifolds in a new way: Given p∈M and q∈S, we
    bound the local feature size (a local version of the reach) of its lifting to
    the tangent space via the inverse exponential map (exp−1p(S)) at q, assuming that
    rchM(S) and the geodesic distance dM(p,q) are bounded. These bounds are motivated
    by the importance of the reach and local feature size to manifold learning, topological
    inference, and triangulating manifolds and the fact that intrinsic approaches
    circumvent the curse of dimensionality.'
acknowledgement: "We thank Eddie Aamari, David Cohen-Steiner, Isa Costantini, Fred
  Chazal, Ramsay Dyer, André Lieutier, and Alef Sterk for discussion and Pierre Pansu
  for encouragement. We further acknowledge the anonymous reviewers whose comments
  helped improve the exposition.\r\nThe research leading to these results has received
  funding from the European Research Council (ERC) under the European Union’s Seventh
  Framework Programme (FP/2007-2013) / ERC Grant Agreement No. 339025 GUDHI (Algorithmic
  Foundations of Geometry Understanding in Higher Dimensions). The first author is
  further supported by the French government, through the 3IA Côte d’Azur Investments
  in the Future project managed by the National Research Agency (ANR) with the reference
  number ANR-19-P3IA-0002. The second author is supported by the European Union’s
  Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie
  Grant Agreement No. 754411 and the Austrian science fund (FWF) M-3073."
article_processing_charge: No
article_type: original
author:
- first_name: Jean Daniel
  full_name: Boissonnat, Jean Daniel
  last_name: Boissonnat
- first_name: Mathijs
  full_name: Wintraecken, Mathijs
  id: 307CFBC8-F248-11E8-B48F-1D18A9856A87
  last_name: Wintraecken
  orcid: 0000-0002-7472-2220
citation:
  ama: Boissonnat JD, Wintraecken M. The reach of subsets of manifolds. <i>Journal
    of Applied and Computational Topology</i>. 2023;7:619-641. doi:<a href="https://doi.org/10.1007/s41468-023-00116-x">10.1007/s41468-023-00116-x</a>
  apa: Boissonnat, J. D., &#38; Wintraecken, M. (2023). The reach of subsets of manifolds.
    <i>Journal of Applied and Computational Topology</i>. Springer Nature. <a href="https://doi.org/10.1007/s41468-023-00116-x">https://doi.org/10.1007/s41468-023-00116-x</a>
  chicago: Boissonnat, Jean Daniel, and Mathijs Wintraecken. “The Reach of Subsets
    of Manifolds.” <i>Journal of Applied and Computational Topology</i>. Springer
    Nature, 2023. <a href="https://doi.org/10.1007/s41468-023-00116-x">https://doi.org/10.1007/s41468-023-00116-x</a>.
  ieee: J. D. Boissonnat and M. Wintraecken, “The reach of subsets of manifolds,”
    <i>Journal of Applied and Computational Topology</i>, vol. 7. Springer Nature,
    pp. 619–641, 2023.
  ista: Boissonnat JD, Wintraecken M. 2023. The reach of subsets of manifolds. Journal
    of Applied and Computational Topology. 7, 619–641.
  mla: Boissonnat, Jean Daniel, and Mathijs Wintraecken. “The Reach of Subsets of
    Manifolds.” <i>Journal of Applied and Computational Topology</i>, vol. 7, Springer
    Nature, 2023, pp. 619–41, doi:<a href="https://doi.org/10.1007/s41468-023-00116-x">10.1007/s41468-023-00116-x</a>.
  short: J.D. Boissonnat, M. Wintraecken, Journal of Applied and Computational Topology
    7 (2023) 619–641.
corr_author: '1'
date_created: 2023-03-26T22:01:08Z
date_published: 2023-09-01T00:00:00Z
date_updated: 2025-04-14T07:44:01Z
day: '01'
department:
- _id: HeEd
doi: 10.1007/s41468-023-00116-x
ec_funded: 1
intvolume: '         7'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://inserm.hal.science/INRIA-SACLAY/hal-04083524v1
month: '09'
oa: 1
oa_version: Submitted Version
page: 619-641
project:
- _id: 260C2330-B435-11E9-9278-68D0E5697425
  call_identifier: H2020
  grant_number: '754411'
  name: ISTplus - Postdoctoral Fellowships
- _id: fc390959-9c52-11eb-aca3-afa58bd282b2
  grant_number: M03073
  name: Learning and triangulating manifolds via collapses
publication: Journal of Applied and Computational Topology
publication_identifier:
  eissn:
  - 2367-1734
  issn:
  - 2367-1726
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
scopus_import: '1'
status: public
title: The reach of subsets of manifolds
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 7
year: '2023'
...