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.
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Csikós, B., Lángi, Z. & Naszódi, M. A generalization of the Discrete Isoperimetric Inequality for Piecewise Smooth Curves of Constant Geodesic Curvature. Period Math Hung 53, 121–131 (2006). https://doi.org/10.1007/s10998-006-0026-z
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DOI: https://doi.org/10.1007/s10998-006-0026-z