What kind of regularity implies the internal ball condition?

Question

Let $\Omega$ be a domain in $\mathbb{R}^n$ with the internal ball condition (at any point $x$ of the bounduary there is a ball contained in $\Omega$ which bounduary contains $x$). What regularity for the bounduary of $\Omega$ dies this imply? I'm pretty sure this is well known, any reference is well appreciated.