Before revision:  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. At the same time, we prove that a careful adaptation of Gonzalez' algorithm in combination with a curve simplification yields a 3-approximation in any dimension, provided that an optimal simplification can be computed exactly.
After revision:  These hardness results hold even when k=1, -hardness extends to the case that \ell=\infty, } i.e., for the minimum-enclosing ball under the Fr\'echet distance. Finally, we observe that a careful adaptation of Gonzalez' algorithm in combination with a curve simplification yields a 3-approximation in any dimension, provided that an optimal simplification can be computed exactly.