A Geometric proof for the Polygonal Isoperimetric Inequality

TL;DR

This paper provides a geometric proof demonstrating that among polygons with a fixed area, the regular polygon minimizes the perimeter, using gradient interpretations and classical triangle geometry.

Contribution

It introduces a geometric approach to prove the polygonal isoperimetric inequality, offering a new perspective compared to traditional methods.

Findings

Regular polygons minimize perimeter for fixed area.

Gradient interpretations facilitate geometric proofs.

The proof relies on classical triangle geometry techniques.

Abstract

Gradients of the perimeter and area of a polygon have straightforward geometric interpretations. The use of optimality conditions for constrained problems and basic ideas in triangle geometry show that polygons with prescribed area minimizing the perimeter must be regular.