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Periodica Mathematica Hungarica

, Volume 53, Issue 1–2, pp 121–131 | Cite as

A generalization of the Discrete Isoperimetric Inequality for Piecewise Smooth Curves of Constant Geodesic Curvature

  • Balázs Csikós
  • Zsolt Lángi
  • Márton Naszódi
Article

Summary

The discrete isoperimetric problem is to determine the maximal area polygon with at most <InlineEquation ID=IE"1"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"2"><EquationSource Format="TEX"><![CDATA[$]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>k$ vertices and of a given perimeter. It is a classical fact that the unique optimal polygon on the Euclidean plane is the regular one. The same statement for the hyperbolic plane was proved by K\'aroly Bezdek [1] and on the sphere by L\'aszl\'o Fejes T\'oth [3]. In the present paper we extend the discrete isoperimetric inequality for ``polygons'' on the three planes of constant curvature bounded by arcs of a given constant geodesic curvature.

hyperbolic plane spherical geometry Euclidean plane isoperimetric inequality 

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Copyright information

© Springer-Verlag/Akadémiai Kiadó 2006

Authors and Affiliations

  • Balázs Csikós
    • 1
  • Zsolt Lángi
    • 2
  • Márton Naszódi
    • 3
  1. 1.Department of Geometry, Eötvös Loránd University
  2. 2.Department of Mathematics and Statistics, University of Calgary
  3. 3.Department of Mathematics and Statistics, University of Calgary

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