First of all let me define what is the the complementary Hausdorff distance between two open sets, I denoted by $d^{H}$ the usual Hausdorff distance in $\mathbb{R}^n$. Let $\Omega_1$ and $\Omega_2$ two open subsets of a (large) compact set $B \subset \mathbb{R}^n$, then their complementary Hau...