To prove an upper bound of 4, I would try this (no guarantee): Start with a square of side 2. Repeatedly lop off a corner to get closer to the polygon. Each time you lop off a corner, you're replacing two sides of a triangle by the third side, so decreasing the circumference. So the circumference is bounded by 8.
The step I'm not sure of is that your original polygon is contained within that square; my intuition on norms is a bit shaky.