---
_id: '10181'
abstract:
- lang: eng
  text: In this article we study some geometric properties of proximally smooth sets.
    First, we introduce a modification of the metric projection and prove its existence.
    Then we provide an algorithm for constructing a rectifiable curve between two
    sufficiently close points of a proximally smooth set in a uniformly convex and
    uniformly smooth Banach space, with the moduli of smoothness and convexity of
    power type. Our algorithm returns a reasonably short curve between two sufficiently
    close points of a proximally smooth set, is iterative and uses our modification
    of the metric projection. We estimate the length of the constructed curve and
    its deviation from the segment with the same endpoints. These estimates coincide
    up to a constant factor with those for the geodesics in a proximally smooth set
    in a Hilbert space.
acknowledgement: Theorem 2 was obtained at Steklov Mathematical Institute RAS and
  supported by Russian Science Foundation, grant N 19-11-00087.
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Grigory
  full_name: Ivanov, Grigory
  id: 87744F66-5C6F-11EA-AFE0-D16B3DDC885E
  last_name: Ivanov
- first_name: Mariana S.
  full_name: Lopushanski, Mariana S.
  last_name: Lopushanski
citation:
  ama: Ivanov G, Lopushanski MS. Rectifiable curves in proximally smooth sets. <i>Set-Valued
    and Variational Analysis</i>. 2021. doi:<a href="https://doi.org/10.1007/s11228-021-00612-1">10.1007/s11228-021-00612-1</a>
  apa: Ivanov, G., &#38; Lopushanski, M. S. (2021). Rectifiable curves in proximally
    smooth sets. <i>Set-Valued and Variational Analysis</i>. Springer Nature. <a href="https://doi.org/10.1007/s11228-021-00612-1">https://doi.org/10.1007/s11228-021-00612-1</a>
  chicago: Ivanov, Grigory, and Mariana S. Lopushanski. “Rectifiable Curves in Proximally
    Smooth Sets.” <i>Set-Valued and Variational Analysis</i>. Springer Nature, 2021.
    <a href="https://doi.org/10.1007/s11228-021-00612-1">https://doi.org/10.1007/s11228-021-00612-1</a>.
  ieee: G. Ivanov and M. S. Lopushanski, “Rectifiable curves in proximally smooth
    sets,” <i>Set-Valued and Variational Analysis</i>. Springer Nature, 2021.
  ista: Ivanov G, Lopushanski MS. 2021. Rectifiable curves in proximally smooth sets.
    Set-Valued and Variational Analysis.
  mla: Ivanov, Grigory, and Mariana S. Lopushanski. “Rectifiable Curves in Proximally
    Smooth Sets.” <i>Set-Valued and Variational Analysis</i>, Springer Nature, 2021,
    doi:<a href="https://doi.org/10.1007/s11228-021-00612-1">10.1007/s11228-021-00612-1</a>.
  short: G. Ivanov, M.S. Lopushanski, Set-Valued and Variational Analysis (2021).
date_created: 2021-10-24T22:01:35Z
date_published: 2021-10-09T00:00:00Z
date_updated: 2023-08-14T08:11:38Z
day: '09'
department:
- _id: UlWa
doi: 10.1007/s11228-021-00612-1
external_id:
  arxiv:
  - '2012.10691'
  isi:
  - '000705774800001'
isi: 1
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://arxiv.org/abs/2012.10691
month: '10'
oa: 1
oa_version: Published Version
publication: Set-Valued and Variational Analysis
publication_identifier:
  eissn:
  - 1877-0541
  issn:
  - 0927-6947
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
scopus_import: '1'
status: public
title: Rectifiable curves in proximally smooth sets
type: journal_article
user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8
year: '2021'
...