New variational arguments regarding the Blaschke-Lebesgue theorem

TL;DR

This paper introduces new variational approaches to the Blaschke-Lebesgue theorem, demonstrating that regular Reuleaux polygons are the only critical points for area, and providing novel proofs for classical geometric theorems.

Contribution

It presents new variational arguments and proofs for the Blaschke-Lebesgue and Firey-Sallee theorems, focusing on the criticality of regular Reuleaux polygons.

Findings

Regular Reuleaux polygons are the only critical points for area.

New variational proofs for classical geometric theorems.

Sensitivity analysis of polygon areas under vertex perturbations.

Abstract

The sensitivity of the areas of Reuleaux polygons and disk polygons is computed with respect to vertex perturbations. Computations are completed for both constrained and Lagrangian formulations and they imply that the only critical Reuleaux polygons for the area functional are the regular ones. As a consequence, new variational proofs for the Blaschke-Lebesgue and Firey-Sallee theorems are found.