---
_id: '9651'
abstract:
- lang: eng
  text: We introduce a hierachy of equivalence relations on the set of separated nets
    of a given Euclidean space, indexed by concave increasing functions ϕ:(0,∞)→(0,∞).
    Two separated nets are called ϕ-displacement equivalent if, roughly speaking,
    there is a bijection between them which, for large radii R, displaces points of
    norm at most R by something of order at most ϕ(R). We show that the spectrum of
    ϕ-displacement equivalence spans from the established notion of bounded displacement
    equivalence, which corresponds to bounded ϕ, to the indiscrete equivalence relation,
    coresponding to ϕ(R)∈Ω(R), in which all separated nets are equivalent. In between
    the two ends of this spectrum, the notions of ϕ-displacement equivalence are shown
    to be pairwise distinct with respect to the asymptotic classes of ϕ(R) for R→∞.
    We further undertake a comparison of our notion of ϕ-displacement equivalence
    with previously studied relations on separated nets. Particular attention is given
    to the interaction of the notions of ϕ-displacement equivalence with that of bilipschitz
    equivalence.
acknowledgement: 'Open access funding provided by Institute of Science and Technology
  (IST Austria). This work was started while both authors were employed at the University
  of Innsbruck and enjoyed the full support of Austrian Science Fund (FWF): P 30902-N35.
  It was continued when the first named author was employed at University of Leipzig
  and the second named author was employed at Institute of Science and Technology
  of Austria, where he was supported by an IST Fellowship.'
article_number: '15'
article_processing_charge: Yes (via OA deal)
article_type: original
arxiv: 1
author:
- first_name: Michael
  full_name: Dymond, Michael
  last_name: Dymond
- first_name: Vojtech
  full_name: Kaluza, Vojtech
  id: 21AE5134-9EAC-11EA-BEA2-D7BD3DDC885E
  last_name: Kaluza
  orcid: 0000-0002-2512-8698
citation:
  ama: Dymond M, Kaluza V. Divergence of separated nets with respect to displacement
    equivalence. <i>Geometriae Dedicata</i>. 2023. doi:<a href="https://doi.org/10.1007/s10711-023-00862-3">10.1007/s10711-023-00862-3</a>
  apa: Dymond, M., &#38; Kaluza, V. (2023). Divergence of separated nets with respect
    to displacement equivalence. <i>Geometriae Dedicata</i>. Springer Nature. <a href="https://doi.org/10.1007/s10711-023-00862-3">https://doi.org/10.1007/s10711-023-00862-3</a>
  chicago: Dymond, Michael, and Vojtech Kaluza. “Divergence of Separated Nets with
    Respect to Displacement Equivalence.” <i>Geometriae Dedicata</i>. Springer Nature,
    2023. <a href="https://doi.org/10.1007/s10711-023-00862-3">https://doi.org/10.1007/s10711-023-00862-3</a>.
  ieee: M. Dymond and V. Kaluza, “Divergence of separated nets with respect to displacement
    equivalence,” <i>Geometriae Dedicata</i>. Springer Nature, 2023.
  ista: Dymond M, Kaluza V. 2023. Divergence of separated nets with respect to displacement
    equivalence. Geometriae Dedicata., 15.
  mla: Dymond, Michael, and Vojtech Kaluza. “Divergence of Separated Nets with Respect
    to Displacement Equivalence.” <i>Geometriae Dedicata</i>, 15, Springer Nature,
    2023, doi:<a href="https://doi.org/10.1007/s10711-023-00862-3">10.1007/s10711-023-00862-3</a>.
  short: M. Dymond, V. Kaluza, Geometriae Dedicata (2023).
date_created: 2021-07-14T07:01:27Z
date_published: 2023-11-17T00:00:00Z
date_updated: 2024-01-11T13:06:32Z
day: '17'
department:
- _id: UlWa
doi: 10.1007/s10711-023-00862-3
external_id:
  arxiv:
  - '2102.13046'
  isi:
  - '001105681500001'
isi: 1
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://doi.org/10.1007/s10711-023-00862-3
month: '11'
oa: 1
oa_version: Published Version
publication: Geometriae Dedicata
publication_identifier:
  eissn:
  - 1572-9168
  issn:
  - 0046-5755
publication_status: epub_ahead
publisher: Springer Nature
quality_controlled: '1'
scopus_import: '1'
status: public
title: Divergence of separated nets with respect to displacement equivalence
type: journal_article
user_id: 3E5EF7F0-F248-11E8-B48F-1D18A9856A87
year: '2023'
...
---
_id: '4080'
abstract:
- lang: eng
  text: This paper proves that any set of n points in the plane contains two points
    such that any circle through those two points encloses at least n12−112+O(1)n47  points
    of the set. The main ingredients used in the proof of this result are edge counting
    formulas for k-order Voronoi diagrams and a lower bound on the minimum number
    of semispaces of size at most k.
acknowledgement: Work on this paper by the first author has been supported by Amoco
  Fnd. Fac. Dev. Comput. Sci. 1-6-44862 and by the National Science Foundation under
  Grant CCR-8714565, by the second author has been partially supported by the Digital
  Equipment Corporation, by the fourth author has been partially supported by the
  Office of Naval Research under Grant N00014-86K-0416.
article_processing_charge: No
article_type: original
author:
- first_name: Herbert
  full_name: Edelsbrunner, Herbert
  id: 3FB178DA-F248-11E8-B48F-1D18A9856A87
  last_name: Edelsbrunner
  orcid: 0000-0002-9823-6833
- first_name: Nany
  full_name: Hasan, Nany
  last_name: Hasan
- first_name: Raimund
  full_name: Seidel, Raimund
  last_name: Seidel
- first_name: Xiao
  full_name: Shen, Xiao
  last_name: Shen
citation:
  ama: Edelsbrunner H, Hasan N, Seidel R, Shen X. Circles through two points that
    always enclose many points. <i>Geometriae Dedicata</i>. 1989;32(1):1-12. doi:<a
    href="https://doi.org/10.1007/BF00181432">10.1007/BF00181432</a>
  apa: Edelsbrunner, H., Hasan, N., Seidel, R., &#38; Shen, X. (1989). Circles through
    two points that always enclose many points. <i>Geometriae Dedicata</i>. Springer.
    <a href="https://doi.org/10.1007/BF00181432">https://doi.org/10.1007/BF00181432</a>
  chicago: Edelsbrunner, Herbert, Nany Hasan, Raimund Seidel, and Xiao Shen. “Circles
    through Two Points That Always Enclose Many Points.” <i>Geometriae Dedicata</i>.
    Springer, 1989. <a href="https://doi.org/10.1007/BF00181432">https://doi.org/10.1007/BF00181432</a>.
  ieee: H. Edelsbrunner, N. Hasan, R. Seidel, and X. Shen, “Circles through two points
    that always enclose many points,” <i>Geometriae Dedicata</i>, vol. 32, no. 1.
    Springer, pp. 1–12, 1989.
  ista: Edelsbrunner H, Hasan N, Seidel R, Shen X. 1989. Circles through two points
    that always enclose many points. Geometriae Dedicata. 32(1), 1–12.
  mla: Edelsbrunner, Herbert, et al. “Circles through Two Points That Always Enclose
    Many Points.” <i>Geometriae Dedicata</i>, vol. 32, no. 1, Springer, 1989, pp.
    1–12, doi:<a href="https://doi.org/10.1007/BF00181432">10.1007/BF00181432</a>.
  short: H. Edelsbrunner, N. Hasan, R. Seidel, X. Shen, Geometriae Dedicata 32 (1989)
    1–12.
date_created: 2018-12-11T12:06:49Z
date_published: 1989-10-01T00:00:00Z
date_updated: 2022-02-14T09:55:28Z
day: '01'
doi: 10.1007/BF00181432
extern: '1'
intvolume: '        32'
issue: '1'
language:
- iso: eng
main_file_link:
- url: https://link.springer.com/article/10.1007/BF00181432
month: '10'
oa_version: None
page: 1 - 12
publication: Geometriae Dedicata
publication_identifier:
  eissn:
  - 1572-9168
  issn:
  - 0046-5755
publication_status: published
publisher: Springer
publist_id: '2043'
quality_controlled: '1'
scopus_import: '1'
status: public
title: Circles through two points that always enclose many points
type: journal_article
user_id: ea97e931-d5af-11eb-85d4-e6957dddbf17
volume: 32
year: '1989'
...