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

Summary The discrete isoperimetric problem is to determine the maximal area polygon with at most ]]>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.