Hausdroff convergence and interior

Question

Let $F_n$ be non empty compact sets of $\mathbb{R^n}$ such that $F_n$ converges to a non empty compact set $F$ in sense of Hausdorff metric,

I would like to know what the conditions are, if there exist, that we must consider so that the interior of $F_n$denoted $\mathring{F_n}$ converges to $\mathring{F}$

Answer 1

A trivial case is when each $\mathring{F_n}$ is dense in $F_n$ and $\mathring{F}$ is dense in $F$, because then Hausdorff distance $d_H$ between $F_n$ and $\mathring{F_n}$ is zero, and $d_H(F,\mathring{F})=0$ too.