Let $E\subset\mathbb{R}^n$ be a bounded domain (open and connected) and $F\Subset E$ be a compactly contained (or even compact) subset of $E$. We set $a := dist(F,\partial E) = \inf\limits_{ y\in F} \inf\limits_{x\in \partial E}|x-y|>0$, where the distance function is taken with respect to the usual euclidean norm $|\cdot|$. Now, if we consider $$G := \Big\{x\in E: dist(x,\partial E)\geq \frac{a}{4}\Big\},$$
then the set $G$ should also be a compactly contained subset of $E$, right? My colleague said that this does not always hold true but I think it should. Thanks in advance for any help!