---
OA_place: publisher
OA_type: gold
_id: '20005'
abstract:
- lang: eng
  text: "We generalize a classical result by Boris Delaunay that introduced Delaunay
    triangulations. In particular, we prove that for a locally finite and coarsely
    dense generic point set A in ℝ^d, every generic point of ℝ^d belongs to exactly
    binom(d+k,d) simplices whose vertices belong to A and whose circumspheres enclose
    exactly k points of A. We extend this result to the cases in which the points
    are weighted, and when A contains only finitely many points in ℝ^d or in \U0001D54A^d.
    Furthermore, we use the result to give a new geometric proof for the fact that
    volumes of hypersimplices are Eulerian numbers."
acknowledgement: "Herbert Edelsbrunner: partially supported by the Wittgenstein Prize,
  Austrian Science\r\nFund (FWF), grant no. Z 342-N31, and by the DFG Collaborative
  Research Center TRR 109,\r\nAustrian Science Fund (FWF), grant no. I 02979-N35.\r\nAlexey
  Garber: partially supported by the Simons Foundation.\r\nMorteza Saghafian: partially
  supported by the Wittgenstein Prize, Austrian Science Fund (FWF),\r\ngrant no. Z
  342-N31, and by the DFG Collaborative Research Center TRR 109, Austrian Science\r\nFund
  (FWF), grant no. I 02979-N35"
alternative_title:
- LIPIcs
article_number: '43'
article_processing_charge: Yes
arxiv: 1
author:
- first_name: Herbert
  full_name: Edelsbrunner, Herbert
  id: 3FB178DA-F248-11E8-B48F-1D18A9856A87
  last_name: Edelsbrunner
  orcid: 0000-0002-9823-6833
- first_name: Alexey
  full_name: Garber, Alexey
  last_name: Garber
- first_name: Morteza
  full_name: Saghafian, Morteza
  id: f86f7148-b140-11ec-9577-95435b8df824
  last_name: Saghafian
citation:
  ama: 'Edelsbrunner H, Garber A, Saghafian M. On spheres with k points inside. In:
    <i>41st International Symposium on Computational Geometry</i>. Vol 332. Schloss
    Dagstuhl - Leibniz-Zentrum für Informatik; 2025. doi:<a href="https://doi.org/10.4230/LIPIcs.SoCG.2025.43">10.4230/LIPIcs.SoCG.2025.43</a>'
  apa: 'Edelsbrunner, H., Garber, A., &#38; Saghafian, M. (2025). On spheres with
    k points inside. In <i>41st International Symposium on Computational Geometry</i>
    (Vol. 332). Kanazawa, Japan: Schloss Dagstuhl - Leibniz-Zentrum für Informatik.
    <a href="https://doi.org/10.4230/LIPIcs.SoCG.2025.43">https://doi.org/10.4230/LIPIcs.SoCG.2025.43</a>'
  chicago: Edelsbrunner, Herbert, Alexey Garber, and Morteza Saghafian. “On Spheres
    with k Points Inside.” In <i>41st International Symposium on Computational Geometry</i>,
    Vol. 332. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2025. <a href="https://doi.org/10.4230/LIPIcs.SoCG.2025.43">https://doi.org/10.4230/LIPIcs.SoCG.2025.43</a>.
  ieee: H. Edelsbrunner, A. Garber, and M. Saghafian, “On spheres with k points inside,”
    in <i>41st International Symposium on Computational Geometry</i>, Kanazawa, Japan,
    2025, vol. 332.
  ista: 'Edelsbrunner H, Garber A, Saghafian M. 2025. On spheres with k points inside.
    41st International Symposium on Computational Geometry. SoCG: Symposium on Computational
    Geometry, LIPIcs, vol. 332, 43.'
  mla: Edelsbrunner, Herbert, et al. “On Spheres with k Points Inside.” <i>41st International
    Symposium on Computational Geometry</i>, vol. 332, 43, Schloss Dagstuhl - Leibniz-Zentrum
    für Informatik, 2025, doi:<a href="https://doi.org/10.4230/LIPIcs.SoCG.2025.43">10.4230/LIPIcs.SoCG.2025.43</a>.
  short: H. Edelsbrunner, A. Garber, M. Saghafian, in:, 41st International Symposium
    on Computational Geometry, Schloss Dagstuhl - Leibniz-Zentrum für Informatik,
    2025.
conference:
  end_date: 2025-06-27
  location: Kanazawa, Japan
  name: 'SoCG: Symposium on Computational Geometry'
  start_date: 2025-06-23
corr_author: '1'
date_created: 2025-07-13T22:01:22Z
date_published: 2025-06-20T00:00:00Z
date_updated: 2025-07-14T07:26:14Z
day: '20'
ddc:
- '510'
department:
- _id: HeEd
doi: 10.4230/LIPIcs.SoCG.2025.43
external_id:
  arxiv:
  - '2410.21204'
file:
- access_level: open_access
  checksum: b5313ed8575ea87913c71a6e3c7513c8
  content_type: application/pdf
  creator: dernst
  date_created: 2025-07-14T07:24:22Z
  date_updated: 2025-07-14T07:24:22Z
  file_id: '20016'
  file_name: 2025_LIPIcs.SoCG_Edelsbrunner.pdf
  file_size: 661893
  relation: main_file
  success: 1
file_date_updated: 2025-07-14T07:24:22Z
fulldoi: https://doi.org/10.4230/LIPIcs.SoCG.2025.43
has_accepted_license: '1'
intvolume: '       332'
language:
- iso: eng
month: '06'
oa: 1
oa_version: Published Version
project:
- _id: 268116B8-B435-11E9-9278-68D0E5697425
  call_identifier: FWF
  grant_number: Z00342
  name: Mathematics, Computer Science
- _id: 2561EBF4-B435-11E9-9278-68D0E5697425
  call_identifier: FWF
  grant_number: I02979-N35
  name: Persistence and stability of geometric complexes
publication: 41st International Symposium on Computational Geometry
publication_identifier:
  eissn:
  - 1868-8969
  isbn:
  - '9783959773706'
publication_status: published
publisher: Schloss Dagstuhl - Leibniz-Zentrum für Informatik
quality_controlled: '1'
scopus_import: '1'
status: public
title: On spheres with k points inside
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: conference
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 332
year: '2025'
...