For smooth regions, both questions were negatively answered. Based on the properties of the ellipse, Penrose and Penrose (1958) constructed a region not illuminable from various points. By modifying this example, Guy and Klee showed how to construct smooth regions not illuminable from any point.
For polygonal bounded regions on the other hand, the solution was not
forthcoming. The problem appeared on various
books of unsolved problems (Klee 1979, Klee and Wagon 1991, Croft and Falconer and Guy 1991). In 1995, Tokarsky
gave a negative answer to the first question by constructing polygonal regions not illuminable from every point.
The second question is still open.