When we are talking about polyominoes, what exacty are the points in the plane we are talking about? Implicit in the problems to this point is that we have been considering polyominoes as closed figures in the plane, that is, figures that include their perimeters as well as their interiors.
What if we consider only the perimeters of polyominoes? It might be possible to come up with succinct covers that are smaller than those of the closed figures. In fact, as you can see to the right, a 12 cell cover is possible. I found this by hand, and haven't proven that it's minimal, but I'm almost certain that it must be.
Conversely, we may consider polyominoes as open figures excluding their perimeters. Surprisingly, sets of these can also have succinct covers smaller than those of "closed" polyominoes. The key is to notice that we can cut out line segments from the interiors of polyominoes we use as pieces of the cover.