Isoperimetric inequality for non-Euclidean polygons

TL;DR

This paper provides a complete proof of the isoperimetric inequality for polygons in hyperbolic and spherical geometries, extending classical Euclidean results to non-Euclidean settings.

Contribution

It offers the first comprehensive proof of the polygonal isoperimetric inequality in hyperbolic and spherical geometries.

Findings

Regular polygons maximize area for given perimeter and sides in non-Euclidean geometries

Complete proof established for hyperbolic and spherical cases

Extends classical Euclidean isoperimetric results to non-Euclidean contexts

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.