Let $P$ be a *simple, closed and bounded* polygon and $p_1,p_2 \in \mathrm{int}(P)$ be two points in its interior. **Is it true that the intersection of the visibility polygons of $p_1$ and $p_2$ is connected?**

The visibility polygon of a point $p\in P$ is the set of all $x \in P$ such that the line segment connecting $p$ and $x$ is contained in $P$.

It might be elementary, but we fail either finding a proof or a counterexample. Furthermore, in the literature I could only find details about the computational aspects of the visibility polygon, but not a single word about its properties. Any reference, example, sketch of proof etc. is welcomed!