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 
 $B\cap V=\emptyset$?



The converse of a Poincaré's result on regular boundary points
The answer is no. Take the unit ball in $R^3$, and remove from it the halfplane
 $x\_3=0, x\_1\geq 0$. This is regular. 


