On the quantitative isoperimetric inequality in the plane with the barycentric asymmetry

In this paper we study a quantitative isoperimetric inequality in the plane, related to the isoperimetric deficit $d$ and the barycentric asymmetry $L_0$. Our aim is to prove that there exists an absolute constant C such that for every planar (convex or compact and connected) set it holds: $L_0^2\le C d$. This generalizes some results obtained by B. Fuglede 1993 in [12]. For that purpose, we consider a shape optimization problem in which we minimize the ratio $d/L_0^2$both in the class of compact connected sets and in the class of convex sets.