This started as a comment, but became too long.

It might be worth mentioning that from a physics point of view, this problem has a certain ambiguity that somewhat diminishes its interest: when the walls are mirrors it is clear what happens when a light ray hits the wall (specular reflection), but what happens when a light ray hits a vertex? The answer to the "illumination problem" depends crucially on how one treats the vertices, because the "dark points" (points inside the region that cannot be illuminated by a point source of light) may appear only if one assumes that a light ray that hits a vertex is extinguished. (Diffuse reflection, rather than specular reflection, might seem a more natural assumption from the physics point of view.)

A variation on this problem that avoids this ambiguity, is to exclude dark points of measure zero, and then ask whether a point source can illuminate the entire interior up to points of measure zero. This is listed as an open problem in <A HREF="http://erikdemaine.org/papers/CCCG2008Open/paper.pdf">the 2008 collection of open problems in computational geometry</A>, by Eric Demaine and our own Joseph O'Rourke.