TITLE: Visibility of vertices in polyhedra
QUESTION [28 upvotes]: Suppose $P$ is a closed polyhedron in space (i.e. a union of polygons which is homeomorphic to $S^2$) and $X$ is an interior point of $P$. Is it true that $X$ can see at least one vertex of $P$? More precisely, does the entire open segment between $X$ and some other vertex lie in the interior of $P$?

REPLY [6 votes]: Here's an orthogonal variation on Ali Khezeli's example, 
which T.S. Michael calls1
the octoplex.
The point at the very center of the polyhedron is not visible to any vertex.

          


          

(Figure by Emmely Rogers.)



1Michael, T.S. How to Guard an Art Gallery and other Discrete Mathematical Adventures. Baltimore: Johns Hopkins U Press, 2009.
  (JHU book link.)