Before revision:  This bound even holds for one-dimensional curves under the continuous Fr\'echet distance , and is further improved to (3\sin(\frac{\pi{3})-\varepsilon) if the curves may lie in the plane and if the }  discrete Fr\'echet distance is used.
After revision:  Our constructions yield different bounds for one and two-dimensional curves and the discrete and continuous Fr\'echet distance , and is further improved to (3\sin(\frac{\pi{3})-\varepsilon) if the curves may lie in the plane and if the } discrete Fr\'echet distance is used.