Does this hold in general?

2017-10-12 03:30:40

I know the following holds when $n=1$. What about the case of $n \geq 2$?

Let $O \subsetneq \mathbb{R}^n$ be a nonempty open set.

There exists $C, \delta >0$ such that $|\{x \in O \mid \operatorname{dist}(x, \mathbb{R} \setminus O) < \varepsilon\}| \geq C {\varepsilon}^n$ for all $0<\varepsilon < \delta$.