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  • 1 Department of Geometry, Eötvös Loránd University Pázmány Péter sétány 1/c, H-1117 Budapest, Hungary Pázmány Péter sétány 1/c, H-1117 Budapest, Hungary
  • 2 Department of Mathematics and Statistics, University of Calgary 2500 University Drive N.W., Calgary, AB, T2N 1N4, Canada 2500 University Drive N.W., Calgary, AB, T2N 1N4, Canada
  • 3 Department of Mathematics and Statistics, University of Calgary 2500 University Drive N.W., Calgary, AB, T2N 1N4, Canada 2500 University Drive N.W., Calgary, AB, T2N 1N4, Canada
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Summary  

The discrete isoperimetric problem is to determine the maximal area polygon with at most \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $k$ \end{document} 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.

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