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. These hardness results hold even when k=1, -hardness extends to the case that \ell=\infty, } i.e., for the  minimum-enclosing ball version of the problem.
After 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 on two-dimensional curves, we show hardness of approximation within a factor close to 2.598. This result also holds when k=1, -hardness extends to the case that \ell=\infty, } i.e., for the minimum-enclosing ball version of the problem.