---
_id: '10222'
abstract:
- lang: eng
  text: Consider a random set of points on the unit sphere in ℝd, which can be either
    uniformly sampled or a Poisson point process. Its convex hull is a random inscribed
    polytope, whose boundary approximates the sphere. We focus on the case d = 3,
    for which there are elementary proofs and fascinating formulas for metric properties.
    In particular, we study the fraction of acute facets, the expected intrinsic volumes,
    the total edge length, and the distance to a fixed point. Finally we generalize
    the results to the ellipsoid with homeoid density.
acknowledgement: "This project has received funding from the European Research Council
  (ERC) under the European Union’s Horizon 2020 research and innovation programme,
  grant no. 788183, from the Wittgenstein Prize, Austrian Science Fund (FWF), grant
  no. Z 342-N31, and from the DFG Collaborative Research Center TRR 109, ‘Discretization
  in Geometry and Dynamics’, Austrian Science Fund (FWF), grant no. I 02979-N35.\r\nWe
  are grateful to Dmitry Zaporozhets and Christoph Thäle for valuable comments and
  for directing us to relevant references. We also thank to Anton Mellit for a useful
  discussion on Bessel functions."
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: Herbert
  full_name: Edelsbrunner, Herbert
  id: 3FB178DA-F248-11E8-B48F-1D18A9856A87
  last_name: Edelsbrunner
  orcid: 0000-0002-9823-6833
- first_name: Anton
  full_name: Nikitenko, Anton
  id: 3E4FF1BA-F248-11E8-B48F-1D18A9856A87
  last_name: Nikitenko
  orcid: 0000-0002-0659-3201
citation:
  ama: Akopyan A, Edelsbrunner H, Nikitenko A. The beauty of random polytopes inscribed
    in the 2-sphere. <i>Experimental Mathematics</i>. 2021:1-15. doi:<a href="https://doi.org/10.1080/10586458.2021.1980459">10.1080/10586458.2021.1980459</a>
  apa: Akopyan, A., Edelsbrunner, H., &#38; Nikitenko, A. (2021). The beauty of random
    polytopes inscribed in the 2-sphere. <i>Experimental Mathematics</i>. Taylor and
    Francis. <a href="https://doi.org/10.1080/10586458.2021.1980459">https://doi.org/10.1080/10586458.2021.1980459</a>
  chicago: Akopyan, Arseniy, Herbert Edelsbrunner, and Anton Nikitenko. “The Beauty
    of Random Polytopes Inscribed in the 2-Sphere.” <i>Experimental Mathematics</i>.
    Taylor and Francis, 2021. <a href="https://doi.org/10.1080/10586458.2021.1980459">https://doi.org/10.1080/10586458.2021.1980459</a>.
  ieee: A. Akopyan, H. Edelsbrunner, and A. Nikitenko, “The beauty of random polytopes
    inscribed in the 2-sphere,” <i>Experimental Mathematics</i>. Taylor and Francis,
    pp. 1–15, 2021.
  ista: Akopyan A, Edelsbrunner H, Nikitenko A. 2021. The beauty of random polytopes
    inscribed in the 2-sphere. Experimental Mathematics., 1–15.
  mla: Akopyan, Arseniy, et al. “The Beauty of Random Polytopes Inscribed in the 2-Sphere.”
    <i>Experimental Mathematics</i>, Taylor and Francis, 2021, pp. 1–15, doi:<a href="https://doi.org/10.1080/10586458.2021.1980459">10.1080/10586458.2021.1980459</a>.
  short: A. Akopyan, H. Edelsbrunner, A. Nikitenko, Experimental Mathematics (2021)
    1–15.
date_created: 2021-11-07T23:01:25Z
date_published: 2021-10-25T00:00:00Z
date_updated: 2023-08-14T11:57:07Z
day: '25'
ddc:
- '510'
department:
- _id: HeEd
doi: 10.1080/10586458.2021.1980459
ec_funded: 1
external_id:
  arxiv:
  - '2007.07783'
  isi:
  - '000710893500001'
file:
- access_level: open_access
  checksum: 3514382e3a1eb87fa6c61ad622874415
  content_type: application/pdf
  creator: dernst
  date_created: 2023-08-14T11:55:10Z
  date_updated: 2023-08-14T11:55:10Z
  file_id: '14053'
  file_name: 2023_ExperimentalMath_Akopyan.pdf
  file_size: 1966019
  relation: main_file
  success: 1
file_date_updated: 2023-08-14T11:55:10Z
has_accepted_license: '1'
isi: 1
language:
- iso: eng
month: '10'
oa: 1
oa_version: Published Version
page: 1-15
project:
- _id: 266A2E9E-B435-11E9-9278-68D0E5697425
  call_identifier: H2020
  grant_number: '788183'
  name: Alpha Shape Theory Extended
- _id: 268116B8-B435-11E9-9278-68D0E5697425
  call_identifier: FWF
  grant_number: Z00342
  name: The Wittgenstein Prize
- _id: 0aa4bc98-070f-11eb-9043-e6fff9c6a316
  grant_number: I4887
  name: Discretization in Geometry and Dynamics
- _id: 2561EBF4-B435-11E9-9278-68D0E5697425
  call_identifier: FWF
  grant_number: I02979-N35
  name: Persistence and stability of geometric complexes
publication: Experimental Mathematics
publication_identifier:
  eissn:
  - 1944-950X
  issn:
  - 1058-6458
publication_status: published
publisher: Taylor and Francis
quality_controlled: '1'
scopus_import: '1'
status: public
title: The beauty of random polytopes inscribed in the 2-sphere
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: 2DF688A6-F248-11E8-B48F-1D18A9856A87
year: '2021'
...