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

Abstract

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.