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

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 $\delta (K)\mathrm{/}{\lambda }_0(K{)}^2$. 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.