A Geometric Solution to the Isoperimetric Problem and its Quantitative Inequalities

TL;DR

This paper introduces a geometric framework to analyze the isoperimetric problem, demonstrating how regular polygons approach the circle in efficiency and extending the analysis to irregular polygons with new quantitative measures.

Contribution

It provides a novel geometric approach to the isoperimetric problem, including efficiency metrics, convergence analysis, and new indices for irregular polygons.

Findings

Regular polygons become more area-efficient as sides increase

Irregular polygons can be analyzed using average apothem and circumradius

New compactness and smoothness indices quantify polygon efficiency

Abstract

This paper presents a geometric approach to the classical isoperimetric problem by analysing the efficiency of regular polygons in enclosing maximum area for a fixed perimeter. Using efficiency metrics, it proves that regular polygons converge to the circle in area efficiency as the number of sides increases. The paper also extends these ideas to irregular polygons using average apothem and circumradius, defines "wasted area," and introduces several compactness and smoothness indices, offering a unified and quantitative view of isoperimetric inequalities