Studia Scientiarum Mathematicarum Hungarica
Volume/Issue:
Volume 61: Issue 3
On the Total Perimeter of Pairwise Disjoint Convex Bodies
Authors:
Arseniy Akopyan FORA Capital, Miami, FL, USA

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and
Alexey Glazyrin School of Mathematical & Statistical Sciences, The University of Texas Rio Grande Valley, Brownsville, TX 78520, USA

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Pages:
237–250
Online Publication Date:
30 Sep 2024
Publication Date:
21 Nov 2024
Article Category:
Research Article
DOI:
https://doi.org/10.1556/012.2024.04317
Keywords:
Convex body; diameter; perimeter; pseudometric; curve of constant width; geometric inequality
Restricted access

In this note we introduce a pseudometric on closed convex planar curves based on distances between normal lines and show its basic properties. Then we use this pseudometric to give a shorter proof of the theorem by Pinchasi that the sum of perimeters of 𝑘 convex planar bodies with disjoint interiors contained in a convex body of perimeter 𝑝 and diameter 𝑑 is not greater than 𝑝 + 2(𝑘 − 1)𝑑.

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    A. Akopyan and R. Karasev. Kadets-Type Theorems for Partitions of a Convex Body. Discrete Comput. Geom., 48:766–776, 2012.

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    A. D. Aleksandrov. Almost everywhere existence of the second differential of a convex function and some properties of convex surfaces connected with it (in Russian). Uch. Zap., Leningrad. Gos. Univ. Math., 6:3–35, 1939.

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    R. D. Anderson and V. L. Klee, Jr. Convex functions and upper semi-continuous collections. Duke Math. J., 19:349–357, 1952.

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    A. Balitskiy. Shortest closed billiard trajectories in the plane and equality cases in Mahler’s conjecture. Geom. Dedicata, 184:121–134, 2016.

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    H. G. Eggleston. Convexity. Cambridge Tracts in Mathematics and Mathematical Physics, 47. Cambridge University Press, New York, 1958.

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    A. Glazyrin and F. Morić. Upper bounds for the perimeter of plane convex bodies. Acta Math. Hungar., 142:366–383, 2014.

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    P. C. Hammer and A. Sobczyk. Planar line families. I. In Proceedings of the American Mathematical Society 4, no. 3, 226–233, 1953.

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    P. C. Hammer and A. Sobczyk. Planar line families. II. In Proceedings of the American Mathematical Society 4, no. 3, 341–349, 1953.

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    H. Martini, L. Montejano and D. Oliveros. Bodies of constant width. Springer International Publishing, 2019.

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    R. Pinchasi. On the perimeter of k pairwise disjoint convex bodies contained in a convex set in the plane. Combinatorica, 37:99–125, 2017.

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    Sh. Tanno. 𝐶∞-approximation of continuous ovals of constant width. J. Math. Soc. Japan, 28(2):384–395, 1976.

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    V. Toponogov. Differential Geometry of Curves and Surfaces. Birkhäuser-Verlag, 2006.

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    B. Wegner. Analytic approximation of continuous ovals of constant width. J. Math. Soc. Japan, 29(3):537–540, 1977.

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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) 

Managing Editor

Gábor SÁGI (Rényi Institute of Mathematics)

Editorial Board

  • Imre BÁRÁNY (Rényi Institute of Mathematics)
  • Károly BÖRÖCZKY (Rényi Institute of Mathematics and Central European University)
  • Péter CSIKVÁRI (ELTE, Budapest) 
  • Joshua GREENE (Boston College)
  • Penny HAXELL (University of Waterloo)
  • Andreas HOLMSEN (Korea Advanced Institute of Science and Technology)
  • Ron HOLZMAN (Technion, Haifa)
  • 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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