The question I'm going to ask is inspired by this thread. I wonder what happens if instead of tilings we consider minimal covers, i.e., families of convex closed polygons that cover the square and from which no polygon can be removed without creating a hole. In that case the key step in my proof of the upper bound fails dramatically (Let $A$ be a regular polygon with a huge number of sides; rotate it a bit to get $B$, then consider small triangles at the "concave" vertices of $A\cup B$. They connect $A$ to $B$, no $3$ can be intersected by a line, and they are as many as one wants). However, I find it rather hard to believe that there is no upper bound in this case. So, can we get some (I don't care how bad) estimate in this case or my intuition is wrong and we can construct arbitrarily large minimal covers? Recall that the condition was that no line should intersect more than $k$ polygons. Also, it would be nice to have a proof that generalizes to higher dimensions (my proof for tilings trivially does) though it is not formally requested.

Any thoughts?