Let $V$ be a bounded open set in $\mathbb{R}^n$ with $n>1$. According to a well known result due to Poincaré, if $x$ is a point in the boundary $\partial V$ and there exists a ball $B$ such that $x\in\partial B$ and $B\cap V=\emptyset$, then $x$ is a regular point for the Dirichlet problem.

Is there a converse for this result? More generally, under which condition(s) can we say that if $x\in\partial V$, there exists a ball $B$ such that $x\in\partial B$ and $B\cap V=\emptyset$?