Given a polyhedron in the usual three dimensional space you can consider a specific point $y$ and define the points which can be "seen" by that point $y$. A point $x$ is seen by $y$ if the line connecting $x$ and $y$ does not pass through a point on the polyhedron (except possibly $x$ and $y$). A set of points $A$ is said to protect a polyhedron if every point that is inside or on the polyhedron can be seen by a point in $A$. It turns out If we have a polygona in two dimensions and let $A$ be the set of vertices then $A$ protects every polygon. It turns out in three dimensions if you take a polyhedron and let $A$ be the set of vertices $A$ might not protect the polyhedron.
My questions are two.
How can we show the in two dimensions the vertices always protect the polygon?
How can we show in three dimensions the vertices may not protect the polyhedron?
Thank you very much, regards.