Optimal sets for the quantitative isoperimetric inequality in the plane with the barycentric distance

TL;DR

This paper investigates the optimal sets in the plane that minimize the ratio of the isoperimetric deficit to the squared barycentric distance, establishing their existence, regularity, and geometric properties.

Contribution

It proves the existence, regularity, and geometric structure of optimal sets for the quantitative isoperimetric inequality involving barycentric distance in the plane.

Findings

Optimal sets have exactly two connected components.

Their boundary contains no circular arcs.

Existence of optimal sets is proven for large enough diameter D.

Abstract

In a recent paper, C. Gambicchia and A. Pratelli proved a quantitative isoperimetric inequality involving the isoperimetric deficit $\delta(K)$ and the barycentric distance $\lambda_0(K)$ for sets $K\subset \mathbb{R}^N$ with given diameter $D$ and measure. In this work we are interested in the optimal sets for this inequality in the plane, i.e. sets that minimize the ratio $\delta(K)/\lambda_0(K)^2$. We prove existence of optimal sets (at least when $D$ is large enough), regularity and express the optimality conditions. Moreover, we prove that the optimal sets have exactly two connected components and their boundary does not contain any arc of circle.