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 ${\mathcal{Q}}_{\ast }$ in ${\mathbb{R}}^3$ involving the Hausdorff asymmetry. We prove that ${\mathcal{Q}}_{\ast }$ attains its minimum over the class $\mathcal{A}$ of convex bodies of fixed volume.