Let $n \geq 2$.
I want to know an example of nonempty open set $O \subsetneq \mathbb{R}^n$ such that :
There exists $C, \delta >0$ such that $|\{x \in O \mid \operatorname{dist}(x, \mathbb{R} \setminus O) < \varepsilon\}| \leq C {\varepsilon}^n$ for all $0<\varepsilon < \delta$,
where $|\cdot|$ indicates the lebesgue measure.
I know the case $O=\mathbb{R}^n \setminus \{x_1, x_2, \cdots, x_m\}$. Does there exist another example?