Steiner's formula and a variational proof of the isoperimetric inequality

TL;DR

This paper presents a novel proof of the planar isoperimetric inequality using Steiner's formula, providing a direct approach that avoids the need to prove the existence of an optimal domain.

Contribution

It introduces a variational proof based on Steiner's formula, simplifying the understanding of the isoperimetric inequality in the plane.

Findings

Provides a direct proof of the isoperimetric inequality

Bypasses the need to establish the existence of an optimal domain

Simplifies the proof structure for the inequality

Abstract

We give a new proof of the isoperimetric inequality in the plane, based on Steiner's formula for the area of a convex neighborhood. This proof establishes the isoperimetric inequality directly, without requiring that we separately establish the existence of an optimal domain. In doing so, this proof bypasses the main difficulty in all of the proofs Steiner outlined for the plane isoperimetric inequality.