Let $\Omega$ be a bounded connected domain in $\mathbb R^n$ with compact smooth boundary. So $\partial \Omega$ can be viewed as a smooth submanifold of dimension n-1. Let $$\Omega_{\epsilon} : = \{ x\in \Omega\ \mid \ dist(x,\partial \Omega) \ge \epsilon\}.$$
I think it is true that there exists $\epsilon_0 >0 $ such that if $0<\epsilon < \epsilon_0$, then $\partial \Omega_{\epsilon}$ is also a smooth submanifold. However I couldn't give a proof for this fact.
Also does it exist an $\epsilon_1 >0$ such that if $0<\epsilon<\epsilon_1$, then the size $ \mid \Omega \setminus \Omega_{\epsilon}\mid \le C\epsilon$, where $C$ is a constant that depends on $\Omega$ only.
I am very interested in these geometric facts and I would appreciate if you can point me any reference on this topic. Thanks a head.