We say that a domain $\Omega \subset \mathbb{R}^n$ satisfies the exterior (interior, resp.) sphere condition if for any $x\in \partial \Omega$, there exist a sphere $B_r(y) \subset \mathbb{R}^n \setminus \Omega$ ($B_r(y) \subset \Omega$, resp.) and $x \in \partial B_r(y)$. If we can take a radius $r>0$ independent of point $x$, we say that $\Omega$ satisfies the uniform sphere condition.
I found that if $\Omega$ is bounded and $\partial \Omega$ is of class $C^{1,1}$ then it satisfies the uniform (exterior and interior) shpere condition, in here (Theorem 1.0.9, page 7) : https://mospace.umsystem.edu/xmlui/bitstream/handle/10355/9670/research.pdf (It states that 'if and only if' but I'm just interested in 'only if' part.)
However, it only gives the sketch of proof so that I couldn't get it. I think that the following answer is more detail even though it proves $C^2$ case.
Smooth boundary condition implies exterior sphere condition
I have two questions in this page :
- Can I take $r$ continuously? If it's true, then $\Omega$ satisfies the uniform sphere condition since $\partial \Omega$ is compact.
- Can I modify it to $C^{1,1}$ case?
