---
_id: '909'
abstract:
- lang: eng
  text: We study the lengths of curves passing through a fixed number of points on
    the boundary of a convex shape in the plane. We show that, for any convex shape
    K, there exist four points on the boundary of K such that the length of any curve
    passing through these points is at least half of the perimeter of K. It is also
    shown that the same statement does not remain valid with the additional constraint
    that the points are extreme points of K. Moreover, the factor ½ cannot
    be achieved with any fixed number of extreme points. We conclude the paper with
    a few other inequalities related to the perimeter of a convex shape.
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: Vladislav
  full_name: Vysotsky, Vladislav
  last_name: Vysotsky
citation:
  ama: Akopyan A, Vysotsky V. On the lengths of curves passing through boundary points
    of a planar convex shape. <i>The American Mathematical Monthly</i>. 2017;124(7):588-596.
    doi:<a href="https://doi.org/10.4169/amer.math.monthly.124.7.588">10.4169/amer.math.monthly.124.7.588</a>
  apa: Akopyan, A., &#38; Vysotsky, V. (2017). On the lengths of curves passing through
    boundary points of a planar convex shape. <i>The American Mathematical Monthly</i>.
    Mathematical Association of America. <a href="https://doi.org/10.4169/amer.math.monthly.124.7.588">https://doi.org/10.4169/amer.math.monthly.124.7.588</a>
  chicago: Akopyan, Arseniy, and Vladislav Vysotsky. “On the Lengths of Curves Passing
    through Boundary Points of a Planar Convex Shape.” <i>The American Mathematical
    Monthly</i>. Mathematical Association of America, 2017. <a href="https://doi.org/10.4169/amer.math.monthly.124.7.588">https://doi.org/10.4169/amer.math.monthly.124.7.588</a>.
  ieee: A. Akopyan and V. Vysotsky, “On the lengths of curves passing through boundary
    points of a planar convex shape,” <i>The American Mathematical Monthly</i>, vol.
    124, no. 7. Mathematical Association of America, pp. 588–596, 2017.
  ista: Akopyan A, Vysotsky V. 2017. On the lengths of curves passing through boundary
    points of a planar convex shape. The American Mathematical Monthly. 124(7), 588–596.
  mla: Akopyan, Arseniy, and Vladislav Vysotsky. “On the Lengths of Curves Passing
    through Boundary Points of a Planar Convex Shape.” <i>The American Mathematical
    Monthly</i>, vol. 124, no. 7, Mathematical Association of America, 2017, pp. 588–96,
    doi:<a href="https://doi.org/10.4169/amer.math.monthly.124.7.588">10.4169/amer.math.monthly.124.7.588</a>.
  short: A. Akopyan, V. Vysotsky, The American Mathematical Monthly 124 (2017) 588–596.
date_created: 2018-12-11T11:49:09Z
date_published: 2017-01-01T00:00:00Z
date_updated: 2023-10-17T11:24:57Z
day: '01'
department:
- _id: HeEd
doi: 10.4169/amer.math.monthly.124.7.588
ec_funded: 1
external_id:
  arxiv:
  - '1605.07997'
  isi:
  - '000413947300002'
intvolume: '       124'
isi: 1
issue: '7'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://arxiv.org/abs/1605.07997
month: '01'
oa: 1
oa_version: Submitted Version
page: 588 - 596
project:
- _id: 25681D80-B435-11E9-9278-68D0E5697425
  call_identifier: FP7
  grant_number: '291734'
  name: International IST Postdoc Fellowship Programme
publication: The American Mathematical Monthly
publication_identifier:
  issn:
  - '00029890'
publication_status: published
publisher: Mathematical Association of America
publist_id: '6534'
quality_controlled: '1'
scopus_import: '1'
status: public
title: On the lengths of curves passing through boundary points of a planar convex
  shape
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 124
year: '2017'
...