Isoperimetric inequality for non-Euclidean polygons

Abstract

It is a classical fact in Euclidean geometry that the regular polygon maximizes area amongst polygons of the same perimeter and number of sides, and the analogue of this in non-Euclidean geometries has long been a folklore result. In this note, we present a complete proof of this polygonal isoperimetric inequality in hyperbolic and spherical geometries.