Covering Polygons Is Hard

We show the following for polygons without holes: 1. covering the interior or boundary of an arbitrary polygon with convex polygons is NP-hard; 2. covering the vertices of an arbitrary polygon with convex polygons is NP-complete; 3. covering the interior or boundary of an orthogonal polygon with rectangles is NP-complete. We note that these results hold even if the polygons are required to be in general position