Studia Scientiarum Mathematicarum Hungarica
Volume/Issue:
Volume 60: Issue 4
The Extreme Polygons for the Self Chebyshev Radius of the Boundary
Authors:
Evgeniĭ Vitalievich Nikitenko Rubtsovsk Industrial Institute of Altai State Technical University after I. I. Polzunov, Rubtsovsk, Traktornaya st., 2/6, 658207, Russia

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and
Yuriĭ Gennadievich Nikonorov Southern Mathematical Institute of the Vladikavkaz Scientific Center of the Russian Academy of Sciences, Vladikavkaz, Vatutina st., 53, 362025, Russia

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Pages:
193–236
Online Publication Date:
29 Jan 2024
Publication Date:
29 Jan 2024
Article Category:
Research Article
DOI:
https://doi.org/10.1556/012.2023.04297
Keywords:
Approximation by polytopes; convex curve; convex polygon; Chebyshev radius; minimum perimeter; self Chebyshev radius
Restricted access

The paper is devoted to some extremal problems for convex polygons on the Euclidean plane, related to the concept of self Chebyshev radius for the polygon boundary. We consider a general problem of minimization of the perimeter among all 𝑛-gons with a fixed self Chebyshev radius of the boundary. The main result of the paper is the complete solution of the mentioned problem for 𝑛 = 4: We proved that the quadrilateral of minimum perimeter is a so called magic kite, that verified the corresponding conjecture by Rolf Walter.

  • [1]

    A. R. Alimov and I. G. Tsar’kov. Chebyshev centres, Jung constants, and their applications. Russ. Math. Surv., 74(5):775–849, 2019.

  • [2]

    C. Alsina and R. B. Nelsen. A cornucopia of quadrilaterals. The Dolciani Mathematical Expositions, 55. MAA Press, Providence, RI; American Mathematical Society, Providence, RI, 2020.

    • Search Google Scholar
    • Export Citation
  • [3]

    D. Amir. Characterizations of Inner Product Spaces. Operator Theory: Advances and Applications Series Profile, Vol. 20. Basel–Boston–Stuttgart: Birkhäuser–Verlag. Basel, 1986.

    • Search Google Scholar
    • Export Citation
  • [4]

    D. Amir and Z. Ziegler. Relative Chebyshev centers in normed linear spaces, I. J. Approximation Theory, 29:235–252, 1980.

  • [5]

    C. Audet, P. Hansen and F. Messine. The small octagon with longest perimeter. J. Combin. Th. A, 114:135–150, 2007.

  • [6]

    C. Audet, P. Hansen and F. Messine. Isoperimetric polygons of maximum width. Discrete Comput. Geom., 4(1):45–60, 2009.

  • [7]

    C. Audet, P. Hansen, F. Messine, and J. Xiong. The largest small octagon. J. Combin. Th. A, 98:46–59, 2002.

  • [8]

    V. Balestro, H. Martini, Yu. G. Nikonorov, and Yu. V. Nikonorova. Extremal problems for convex curves with a given self Chebyshev radius. Results in Mathematics, 76(2), 2021, Paper No. 87, 13 pp.

    • Search Google Scholar
    • Export Citation
  • [9]

    H. H. Bauschke and P. K. Combettes. Convex analysis and monotone operator theory in Hilbert spaces, second edition. Cham: Springer, XIX+619 p., 2017.

    • Search Google Scholar
    • Export Citation
  • [10]

    T. Bonnesen and W. Fenchel. Theory of Convex Bodies, BCS Associates, Moscow, ID, 1987. Translated from the German and edited by L. Boron, C. Christenson and B. Smith.

    • Search Google Scholar
    • Export Citation
  • [11]

    K. J. Falconer. A characterisation of plane curves of constant width. J. Lond. Math. Soc., II. Ser., 16:536–538, 1977.

  • [12]

    R. Fedorov, A. Belov, A. Kovaldzhi, I. Yashchenko, and S. Levy (editors). Moscow Mathematical Olympiads, 1993–1999. Translation of the 2006 Russian original. MSRI Mathematical Circles Library, 4. Providence, RI: American Mathematical Society (AMS); Berkeley, CA: Mathematical Sciences Research Institute (MSRI), 2011.

    • Search Google Scholar
    • Export Citation
  • [13]

    R. Graham. The largest small hexagon. J. Combin. Th. A, 18:165–170, 1975.

  • [14]

    K. H. Hang and H. Wang. Solving problems in geometry. Insights and strategies. Mathematical Olympiad Series 10. Hackensack, NJ: World Scientific, 2017.

    • Search Google Scholar
    • Export Citation
  • [15]

    K. G. Hare and M. J. Mossinghoff. Most Reinhardt polygons are sporadic. Geom. Dedicata, 198:1–18, 2019.

  • [16]

    R. H. Landau. A first course in scientific computing. Symbolic, graphic, and numeric modeling using Maple, Java, Mathematica, and Fortran90. With contributions by R. Wangberg, K. Augustson, M. J. Paez, C. C. Bordeianu and C. Barnes. Princeton University Press, Princeton, NJ, 2005.

    • Search Google Scholar
    • Export Citation
  • [17]

    H. Martini, L. Montejano, and D. Oliveros. Bodies of Constant Width. An Introduction to Convex Geometry with Applications. Birkhäuser/Springer, Cham, 2019.

    • Search Google Scholar
    • Export Citation
  • [18]

    V. V. Prasolov. Problems in Plane Geometry, 5th ed., rev. and compl. (Russian). Moscow, Russia: The Moscow Center for Continuous Mathematical Education, 2006. Online access: https://mccme.ru/free-books/prasolov/planim5.pdf

    • Search Google Scholar
    • Export Citation
  • [19]

    K. Reinhardt. Extremale Polygone gegebenen Durchmessers. Jahresbericht der Deutschen Mathematiker-Vereinigung, 31:251–270, 1922.

  • [20]

    R. Walter. On a minimax problem for ovals. Minimax Theory Appl. 2(2):285–318, 2017. See also arXiv:1606.06717.

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Cover Studia Scientiarum Mathematicarum Hungarica
Studia Scientiarum Mathematicarum Hungarica
Print ISSN:
0081-6906
Online ISSN:
1588-2896

Editors in Chief

Gábor SIMONYI (Rényi Institute of Mathematics)
András STIPSICZ (Rényi Institute of Mathematics)
Géza TÓTH (Rényi Institute of Mathematics) 

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Gábor SÁGI (Rényi Institute of Mathematics)

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  • Károly BÖRÖCZKY (Rényi Institute of Mathematics and Central European University)
  • Péter CSIKVÁRI (ELTE, Budapest) 
  • Joshua GREENE (Boston College)
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  • Andreas HOLMSEN (Korea Advanced Institute of Science and Technology)
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  • Satoru IWATA (University of Tokyo)
  • Tibor JORDÁN (ELTE, Budapest)
  • Roy MESHULAM (Technion, Haifa)
  • Frédéric MEUNIER (École des Ponts ParisTech)
  • Márton NASZÓDI (ELTE, Budapest)
  • Eran NEVO (Hebrew University of Jerusalem)
  • János PACH (Rényi Institute of Mathematics)
  • Péter Pál PACH (BME, Budapest)
  • Andrew SUK (University of California, San Diego)
  • Zoltán SZABÓ (Princeton University)
  • Martin TANCER (Charles University, Prague)
  • Gábor TARDOS (Rényi Institute of Mathematics)
  • Paul WOLLAN (University of Rome "La Sapienza")

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Studia Scientiarum Mathematicarum Hungarica
Language English
French
German
Size B5
Year of
Foundation		 1966
Volumes
per Year		 1
Issues
per Year		 4
Founder Magyar Tudományos Akadémia  
Founder's
Address		 H-1051 Budapest, Hungary, Széchenyi István tér 9.
Publisher Akadémiai Kiadó
Publisher's
Address		 H-1117 Budapest, Hungary 1516 Budapest, PO Box 245.
Responsible
Publisher		 Chief Executive Officer, Akadémiai Kiadó
ISSN 0081-6906 (Print)
ISSN 1588-2896 (Online)

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