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.
After revision:  These hardness results hold even when k=1, and the \mathsf{NP-hardness extends to the case that \ell=\infty, } i.e., for the minimum-enclosing ball version of the problem.