  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 $E\subset \mathbb{R}^2$.
