A note on existence of an optimal set for a bonnesen type quantitative isoperimetric ratio in the plane

In this note we prove the existence of a set $E_0\subset\mathbb{R}^2$, different from a ball, which minimizes, among the convex sets that satisfy a suitable interior cone condition, the ratio \begin{equation} \label{eq:0} \frac{D(E)}{\lambda_\mathcal{H}^2(E)}, \end{equation} where $D$ is the isoperimetric deficit and $\lambda_\mathcal{H}$ the deviation from the spherical shape of a set $E\subset \mathbb{R}^2$.