---
_id: '14660'
abstract:
- lang: eng
  text: "The classical Steinitz theorem states that if the origin belongs to the interior
    of the convex hull of a set \U0001D446⊂ℝ\U0001D451, then there are at most 2\U0001D451
    points of \U0001D446 whose convex hull contains the origin in the interior. Bárány,
    Katchalski,and Pach proved the following quantitative version of Steinitz’s theorem.
    Let \U0001D444 be a convex polytope in ℝ\U0001D451 containing the standard Euclidean
    unit ball \U0001D401\U0001D451. Then there exist at most 2\U0001D451 vertices
    of \U0001D444 whose convex hull \U0001D444′ satisfies \U0001D45F\U0001D401\U0001D451⊂\U0001D444′
    with \U0001D45F⩾\U0001D451−2\U0001D451. They conjectured that \U0001D45F⩾\U0001D450\U0001D451−1∕2
    holds with a universal constant \U0001D450>0. We prove \U0001D45F⩾15\U0001D4512,
    the first polynomial lower bound on \U0001D45F. Furthermore, we show that \U0001D45F
    is not greater than 2/√\U0001D451."
acknowledgement: M.N. was supported by the János Bolyai Scholarship of the Hungarian
  Academy of Sciences aswell as the National Research, Development and Innovation
  Fund (NRDI) grants K119670 andK131529, and the ÚNKP-22-5 New National Excellence
  Program of the Ministry for Innovationand Technology from the source of the NRDI
  as well as the ELTE TKP 2021-NKTA-62 fundingscheme
article_processing_charge: Yes (via OA deal)
article_type: original
arxiv: 1
author:
- first_name: Grigory
  full_name: Ivanov, Grigory
  id: 87744F66-5C6F-11EA-AFE0-D16B3DDC885E
  last_name: Ivanov
- first_name: Márton
  full_name: Naszódi, Márton
  last_name: Naszódi
citation:
  ama: 'Ivanov G, Naszódi M. Quantitative Steinitz theorem: A polynomial bound. <i>Bulletin
    of the London Mathematical Society</i>. 2024;56(2):796-802. doi:<a href="https://doi.org/10.1112/blms.12965">10.1112/blms.12965</a>'
  apa: 'Ivanov, G., &#38; Naszódi, M. (2024). Quantitative Steinitz theorem: A polynomial
    bound. <i>Bulletin of the London Mathematical Society</i>. London Mathematical
    Society. <a href="https://doi.org/10.1112/blms.12965">https://doi.org/10.1112/blms.12965</a>'
  chicago: 'Ivanov, Grigory, and Márton Naszódi. “Quantitative Steinitz Theorem: A
    Polynomial Bound.” <i>Bulletin of the London Mathematical Society</i>. London
    Mathematical Society, 2024. <a href="https://doi.org/10.1112/blms.12965">https://doi.org/10.1112/blms.12965</a>.'
  ieee: 'G. Ivanov and M. Naszódi, “Quantitative Steinitz theorem: A polynomial bound,”
    <i>Bulletin of the London Mathematical Society</i>, vol. 56, no. 2. London Mathematical
    Society, pp. 796–802, 2024.'
  ista: 'Ivanov G, Naszódi M. 2024. Quantitative Steinitz theorem: A polynomial bound.
    Bulletin of the London Mathematical Society. 56(2), 796–802.'
  mla: 'Ivanov, Grigory, and Márton Naszódi. “Quantitative Steinitz Theorem: A Polynomial
    Bound.” <i>Bulletin of the London Mathematical Society</i>, vol. 56, no. 2, London
    Mathematical Society, 2024, pp. 796–802, doi:<a href="https://doi.org/10.1112/blms.12965">10.1112/blms.12965</a>.'
  short: G. Ivanov, M. Naszódi, Bulletin of the London Mathematical Society 56 (2024)
    796–802.
corr_author: '1'
date_created: 2023-12-10T23:00:58Z
date_published: 2024-02-01T00:00:00Z
date_updated: 2025-09-04T11:31:49Z
day: '01'
ddc:
- '510'
department:
- _id: UlWa
doi: 10.1112/blms.12965
external_id:
  arxiv:
  - '2212.04308'
  isi:
  - '001113277100001'
file:
- access_level: open_access
  checksum: 30ea0694757bc668cf7cd15ae357b35e
  content_type: application/pdf
  creator: dernst
  date_created: 2024-07-16T10:35:10Z
  date_updated: 2024-07-16T10:35:10Z
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  file_name: 2024_BulletinLondonMathSoc_Ivanov.pdf
  file_size: 111756
  relation: main_file
  success: 1
file_date_updated: 2024-07-16T10:35:10Z
has_accepted_license: '1'
intvolume: '        56'
isi: 1
issue: '2'
language:
- iso: eng
month: '02'
oa: 1
oa_version: Published Version
page: 796-802
publication: Bulletin of the London Mathematical Society
publication_identifier:
  eissn:
  - 1469-2120
  issn:
  - 0024-6093
publication_status: published
publisher: London Mathematical Society
quality_controlled: '1'
scopus_import: '1'
status: public
title: 'Quantitative Steinitz theorem: A polynomial bound'
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: 317138e5-6ab7-11ef-aa6d-ffef3953e345
volume: 56
year: '2024'
...