On The Quantitative Isoperimetric Inequality In The Plane

TL;DR

This paper investigates the quantitative isoperimetric inequality in the plane, proving the existence of a minimal set different from a ball, and providing new insights into the properties of optimal sets.

Contribution

It establishes the existence of a non-ball set minimizing the isoperimetric deficit to asymmetry ratio and offers a new proof of the inequality.

Findings

Existence of a minimal set different from a ball

New properties of the optimal set

A novel proof of the quantitative isoperimetric inequality

Abstract

In this paper we study the quantitative isoperimetric inequality in the plane. We prove the existence of a set $\Omega$, different from a ball, which minimizes the ratio $\delta(\Omega)/\lambda^2(\Omega)$, where $\delta$ is the isoperimetric deficit and $\lambda$ the Fraenkel asymmetry, giving a new proof ofthe quantitative isoperimetric inequality. Some new properties of the optimal set are also shown.