---
_id: '10608'
abstract:
- lang: eng
  text: We consider infinite-dimensional properties in coarse geometry for hyperspaces
    consisting of finite subsets of metric spaces with the Hausdorff metric. We see
    that several infinite-dimensional properties are preserved by taking the hyperspace
    of subsets with at most n points. On the other hand, we prove that, if a metric
    space contains a sequence of long intervals coarsely, then its hyperspace of finite
    subsets is not coarsely embeddable into any uniformly convex Banach space. As
    a corollary, the hyperspace of finite subsets of the real line is not coarsely
    embeddable into any uniformly convex Banach space. It is also shown that every
    (not necessarily bounded geometry) metric space with straight finite decomposition
    complexity has metric sparsification property.
acknowledgement: We would like to thank the referees for their careful reading and
  the comments that improved our work. The third named author would like to thank
  the Division of Mathematics, Physics and Earth Sciences of the Graduate School of
  Science and Engineering of Ehime University and the second named author for hosting
  his visit in June 2018. Open access funding provided by Institute of Science and
  Technology (IST Austria).
article_processing_charge: Yes (via OA deal)
article_type: original
author:
- first_name: Thomas
  full_name: Weighill, Thomas
  last_name: Weighill
- first_name: Takamitsu
  full_name: Yamauchi, Takamitsu
  last_name: Yamauchi
- first_name: Nicolò
  full_name: Zava, Nicolò
  id: c8b3499c-7a77-11eb-b046-aa368cbbf2ad
  last_name: Zava
  orcid: 0000-0001-8686-1888
citation:
  ama: Weighill T, Yamauchi T, Zava N. Coarse infinite-dimensionality of hyperspaces
    of finite subsets. <i>European Journal of Mathematics</i>. 2022;8(1):335-355.
    doi:<a href="https://doi.org/10.1007/s40879-021-00515-3">10.1007/s40879-021-00515-3</a>
  apa: Weighill, T., Yamauchi, T., &#38; Zava, N. (2022). Coarse infinite-dimensionality
    of hyperspaces of finite subsets. <i>European Journal of Mathematics</i>. Springer
    Nature. <a href="https://doi.org/10.1007/s40879-021-00515-3">https://doi.org/10.1007/s40879-021-00515-3</a>
  chicago: Weighill, Thomas, Takamitsu Yamauchi, and Nicolò Zava. “Coarse Infinite-Dimensionality
    of Hyperspaces of Finite Subsets.” <i>European Journal of Mathematics</i>. Springer
    Nature, 2022. <a href="https://doi.org/10.1007/s40879-021-00515-3">https://doi.org/10.1007/s40879-021-00515-3</a>.
  ieee: T. Weighill, T. Yamauchi, and N. Zava, “Coarse infinite-dimensionality of
    hyperspaces of finite subsets,” <i>European Journal of Mathematics</i>, vol. 8,
    no. 1. Springer Nature, pp. 335–355, 2022.
  ista: Weighill T, Yamauchi T, Zava N. 2022. Coarse infinite-dimensionality of hyperspaces
    of finite subsets. European Journal of Mathematics. 8(1), 335–355.
  mla: Weighill, Thomas, et al. “Coarse Infinite-Dimensionality of Hyperspaces of
    Finite Subsets.” <i>European Journal of Mathematics</i>, vol. 8, no. 1, Springer
    Nature, 2022, pp. 335–55, doi:<a href="https://doi.org/10.1007/s40879-021-00515-3">10.1007/s40879-021-00515-3</a>.
  short: T. Weighill, T. Yamauchi, N. Zava, European Journal of Mathematics 8 (2022)
    335–355.
date_created: 2022-01-09T23:01:27Z
date_published: 2022-03-01T00:00:00Z
date_updated: 2024-05-22T11:10:22Z
day: '01'
ddc:
- '500'
department:
- _id: HeEd
doi: 10.1007/s40879-021-00515-3
file:
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  checksum: ce35cbb2d8c889dc7750719972634ed4
  content_type: application/pdf
  creator: kschuh
  date_created: 2024-05-22T11:10:10Z
  date_updated: 2024-05-22T11:10:10Z
  file_id: '17036'
  file_name: 2022_EuJournalMath_Weighill.pdf
  file_size: 371515
  relation: main_file
  success: 1
file_date_updated: 2024-05-22T11:10:10Z
has_accepted_license: '1'
intvolume: '         8'
issue: '1'
language:
- iso: eng
month: '03'
oa: 1
oa_version: Published Version
page: 335-355
publication: European Journal of Mathematics
publication_identifier:
  eissn:
  - 2199-6768
  issn:
  - 2199-675X
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
scopus_import: '1'
status: public
title: Coarse infinite-dimensionality of hyperspaces of finite subsets
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: 3E5EF7F0-F248-11E8-B48F-1D18A9856A87
volume: 8
year: '2022'
...
---
_id: '7791'
abstract:
- lang: eng
  text: Extending a result of Milena Radnovic and Serge Tabachnikov, we establish
    conditionsfor two different non-symmetric norms to define the same billiard reflection
    law.
acknowledgement: AA was supported by European Research Council (ERC) under the European
  Union’s Horizon 2020 research and innovation programme (Grant Agreement No. 78818
  Alpha). RK was supported by the Federal professorship program Grant 1.456.2016/1.4
  and the Russian Foundation for Basic Research Grants 18-01-00036 and 19-01-00169.
  Open access funding provided by Institute of Science and Technology (IST Austria).
  The authors thank Alexey Balitskiy, Milena Radnović, and Serge Tabachnikov for useful
  discussions.
article_processing_charge: Yes (via OA deal)
article_type: original
arxiv: 1
author:
- first_name: Arseniy
  full_name: Akopyan, Arseniy
  id: 430D2C90-F248-11E8-B48F-1D18A9856A87
  last_name: Akopyan
  orcid: 0000-0002-2548-617X
- first_name: Roman
  full_name: Karasev, Roman
  last_name: Karasev
citation:
  ama: Akopyan A, Karasev R. When different norms lead to same billiard trajectories?
    <i>European Journal of Mathematics</i>. 2022;8(4):1309-1312. doi:<a href="https://doi.org/10.1007/s40879-020-00405-0">10.1007/s40879-020-00405-0</a>
  apa: Akopyan, A., &#38; Karasev, R. (2022). When different norms lead to same billiard
    trajectories? <i>European Journal of Mathematics</i>. Springer Nature. <a href="https://doi.org/10.1007/s40879-020-00405-0">https://doi.org/10.1007/s40879-020-00405-0</a>
  chicago: Akopyan, Arseniy, and Roman Karasev. “When Different Norms Lead to Same
    Billiard Trajectories?” <i>European Journal of Mathematics</i>. Springer Nature,
    2022. <a href="https://doi.org/10.1007/s40879-020-00405-0">https://doi.org/10.1007/s40879-020-00405-0</a>.
  ieee: A. Akopyan and R. Karasev, “When different norms lead to same billiard trajectories?,”
    <i>European Journal of Mathematics</i>, vol. 8, no. 4. Springer Nature, pp. 1309–1312,
    2022.
  ista: Akopyan A, Karasev R. 2022. When different norms lead to same billiard trajectories?
    European Journal of Mathematics. 8(4), 1309–1312.
  mla: Akopyan, Arseniy, and Roman Karasev. “When Different Norms Lead to Same Billiard
    Trajectories?” <i>European Journal of Mathematics</i>, vol. 8, no. 4, Springer
    Nature, 2022, pp. 1309–12, doi:<a href="https://doi.org/10.1007/s40879-020-00405-0">10.1007/s40879-020-00405-0</a>.
  short: A. Akopyan, R. Karasev, European Journal of Mathematics 8 (2022) 1309–1312.
corr_author: '1'
date_created: 2020-05-03T22:00:48Z
date_published: 2022-12-01T00:00:00Z
date_updated: 2025-04-14T07:48:36Z
day: '01'
ddc:
- '510'
department:
- _id: HeEd
doi: 10.1007/s40879-020-00405-0
ec_funded: 1
external_id:
  arxiv:
  - '1912.12685'
file:
- access_level: open_access
  checksum: f53e71fd03744075adcd0b8fc1b8423d
  content_type: application/pdf
  creator: dernst
  date_created: 2020-05-04T10:33:42Z
  date_updated: 2020-07-14T12:48:03Z
  file_id: '7796'
  file_name: 2020_EuropMathematics_Akopyan.pdf
  file_size: 263926
  relation: main_file
file_date_updated: 2020-07-14T12:48:03Z
has_accepted_license: '1'
intvolume: '         8'
issue: '4'
language:
- iso: eng
month: '12'
oa: 1
oa_version: Published Version
page: 1309 - 1312
project:
- _id: 266A2E9E-B435-11E9-9278-68D0E5697425
  call_identifier: H2020
  grant_number: '788183'
  name: Alpha Shape Theory Extended
- _id: B67AFEDC-15C9-11EA-A837-991A96BB2854
  name: IST Austria Open Access Fund
publication: European Journal of Mathematics
publication_identifier:
  eissn:
  - 2199-6768
  issn:
  - 2199-675X
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
scopus_import: '1'
status: public
title: When different norms lead to same billiard trajectories?
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: 3E5EF7F0-F248-11E8-B48F-1D18A9856A87
volume: 8
year: '2022'
...
---
_id: '8538'
abstract:
- lang: eng
  text: We prove some recent experimental observations of Dan Reznik concerning periodic
    billiard orbits in ellipses. For example, the sum of cosines of the angles of
    a periodic billiard polygon remains constant in the 1-parameter family of such
    polygons (that exist due to the Poncelet porism). In our proofs, we use geometric
    and complex analytic methods.
acknowledgement: " This paper would not be written if not for Dan Reznik’s curiosity
  and persistence; we are very grateful to him. We also thank R. Garcia and J. Koiller
  for interesting discussions. It is a pleasure to thank the Mathematical Institute
  of the University of Heidelberg for its stimulating atmosphere. ST thanks M. Bialy
  for interesting discussions and the Tel Aviv\r\nUniversity for its invariable hospitality.
  AA was supported by European Research Council (ERC) under the European Union’s Horizon
  2020 research and innovation programme (grant agreement No 78818 Alpha). RS is supported
  by NSF Grant DMS-1807320. ST was supported by NSF grant DMS-1510055 and SFB/TRR
  191."
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Arseniy
  full_name: Akopyan, Arseniy
  id: 430D2C90-F248-11E8-B48F-1D18A9856A87
  last_name: Akopyan
  orcid: 0000-0002-2548-617X
- first_name: Richard
  full_name: Schwartz, Richard
  last_name: Schwartz
- first_name: Serge
  full_name: Tabachnikov, Serge
  last_name: Tabachnikov
citation:
  ama: Akopyan A, Schwartz R, Tabachnikov S. Billiards in ellipses revisited. <i>European
    Journal of Mathematics</i>. 2022;8(4):1313-1327. doi:<a href="https://doi.org/10.1007/s40879-020-00426-9">10.1007/s40879-020-00426-9</a>
  apa: Akopyan, A., Schwartz, R., &#38; Tabachnikov, S. (2022). Billiards in ellipses
    revisited. <i>European Journal of Mathematics</i>. Springer Nature. <a href="https://doi.org/10.1007/s40879-020-00426-9">https://doi.org/10.1007/s40879-020-00426-9</a>
  chicago: Akopyan, Arseniy, Richard Schwartz, and Serge Tabachnikov. “Billiards in
    Ellipses Revisited.” <i>European Journal of Mathematics</i>. Springer Nature,
    2022. <a href="https://doi.org/10.1007/s40879-020-00426-9">https://doi.org/10.1007/s40879-020-00426-9</a>.
  ieee: A. Akopyan, R. Schwartz, and S. Tabachnikov, “Billiards in ellipses revisited,”
    <i>European Journal of Mathematics</i>, vol. 8, no. 4. Springer Nature, pp. 1313–1327,
    2022.
  ista: Akopyan A, Schwartz R, Tabachnikov S. 2022. Billiards in ellipses revisited.
    European Journal of Mathematics. 8(4), 1313–1327.
  mla: Akopyan, Arseniy, et al. “Billiards in Ellipses Revisited.” <i>European Journal
    of Mathematics</i>, vol. 8, no. 4, Springer Nature, 2022, pp. 1313–27, doi:<a
    href="https://doi.org/10.1007/s40879-020-00426-9">10.1007/s40879-020-00426-9</a>.
  short: A. Akopyan, R. Schwartz, S. Tabachnikov, European Journal of Mathematics
    8 (2022) 1313–1327.
date_created: 2020-09-20T22:01:38Z
date_published: 2022-12-01T00:00:00Z
date_updated: 2025-04-14T07:48:34Z
day: '01'
department:
- _id: HeEd
doi: 10.1007/s40879-020-00426-9
ec_funded: 1
external_id:
  arxiv:
  - '2001.02934'
intvolume: '         8'
issue: '4'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://arxiv.org/abs/2001.02934
month: '12'
oa: 1
oa_version: Preprint
page: 1313-1327
project:
- _id: 266A2E9E-B435-11E9-9278-68D0E5697425
  call_identifier: H2020
  grant_number: '788183'
  name: Alpha Shape Theory Extended
publication: European Journal of Mathematics
publication_identifier:
  eissn:
  - 2199-6768
  issn:
  - 2199-675X
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
scopus_import: '1'
status: public
title: Billiards in ellipses revisited
type: journal_article
user_id: 3E5EF7F0-F248-11E8-B48F-1D18A9856A87
volume: 8
year: '2022'
...
---
_id: '441'
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Nikita
  full_name: Kalinin, Nikita
  last_name: Kalinin
- first_name: Mikhail
  full_name: Shkolnikov, Mikhail
  id: 35084A62-F248-11E8-B48F-1D18A9856A87
  last_name: Shkolnikov
  orcid: 0000-0002-4310-178X
citation:
  ama: Kalinin N, Shkolnikov M. Tropical formulae for summation over a part of SL(2,Z).
    <i>European Journal of Mathematics</i>. 2019;5(3):909–928. doi:<a href="https://doi.org/10.1007/s40879-018-0218-0">10.1007/s40879-018-0218-0</a>
  apa: Kalinin, N., &#38; Shkolnikov, M. (2019). Tropical formulae for summation over
    a part of SL(2,Z). <i>European Journal of Mathematics</i>. Springer Nature. <a
    href="https://doi.org/10.1007/s40879-018-0218-0">https://doi.org/10.1007/s40879-018-0218-0</a>
  chicago: Kalinin, Nikita, and Mikhail Shkolnikov. “Tropical Formulae for Summation
    over a Part of SL(2,Z).” <i>European Journal of Mathematics</i>. Springer Nature,
    2019. <a href="https://doi.org/10.1007/s40879-018-0218-0">https://doi.org/10.1007/s40879-018-0218-0</a>.
  ieee: N. Kalinin and M. Shkolnikov, “Tropical formulae for summation over a part
    of SL(2,Z),” <i>European Journal of Mathematics</i>, vol. 5, no. 3. Springer Nature,
    pp. 909–928, 2019.
  ista: Kalinin N, Shkolnikov M. 2019. Tropical formulae for summation over a part
    of SL(2,Z). European Journal of Mathematics. 5(3), 909–928.
  mla: Kalinin, Nikita, and Mikhail Shkolnikov. “Tropical Formulae for Summation over
    a Part of SL(2,Z).” <i>European Journal of Mathematics</i>, vol. 5, no. 3, Springer
    Nature, 2019, pp. 909–928, doi:<a href="https://doi.org/10.1007/s40879-018-0218-0">10.1007/s40879-018-0218-0</a>.
  short: N. Kalinin, M. Shkolnikov, European Journal of Mathematics 5 (2019) 909–928.
date_created: 2018-12-11T11:46:29Z
date_published: 2019-09-15T00:00:00Z
date_updated: 2021-01-12T07:56:46Z
day: '15'
department:
- _id: TaHa
doi: 10.1007/s40879-018-0218-0
ec_funded: 1
external_id:
  arxiv:
  - '1711.02089'
intvolume: '         5'
issue: '3'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://arxiv.org/abs/1711.02089
month: '09'
oa: 1
oa_version: Preprint
page: 909–928
project:
- _id: 25681D80-B435-11E9-9278-68D0E5697425
  call_identifier: FP7
  grant_number: '291734'
  name: International IST Postdoc Fellowship Programme
publication: European Journal of Mathematics
publication_identifier:
  eissn:
  - 2199-6768
  issn:
  - 2199-675X
publication_status: published
publisher: Springer Nature
publist_id: '7382'
quality_controlled: '1'
scopus_import: 1
status: public
title: Tropical formulae for summation over a part of SL(2,Z)
type: journal_article
user_id: D865714E-FA4E-11E9-B85B-F5C5E5697425
volume: 5
year: '2019'
...