An existence result for a quantitative isoperimetric inequality in mathbb{R}³ involving the Hausdorff asymmetry

We study the existence of an optimizer for a quantitative isoperimetric ratio $\mathcal{Q}_*$ in $\mathbb{R}^3$ involving the Hausdorff asymmetry. We prove that $\mathcal{Q}_*$ attains its minimum over the class $\mathcal{A}$ of convex bodies of fixed volume.