I found the following problem:
Is it possible to partition every convex polygon into a finite number concave quadrilaterals?
The answer seems negative, because heuristically if we remove a concave quadrilateral the new polygon is still convex, and after a finite number of steps we arrive at a concave quadrilateral in end, and therefore a contradiction.
The problem is that it is possible to have some weird configurations, and removing a any quadrilateral from the partition may make the resulting polygon non-convex.
What is the answer to the question, and what is the proof?
Moreover, is there a more general result like:
It is impossible to partition a convex set into a finite number of regular, connected non-convex sets? (Answer: NO)
What happens if we remove the finiteness assumption?