A maximization problem

Consider the minimization problem described this paper. Let $ f_{\lambda}$ be the minimizer. As a part of extending my work, I am able to show the following facts

$ $ \lim_\limits{\lambda \to 0}\|f_{\lambda}\| = 0$ $ and $ $ \lim_\limits{\lambda \to \infty}\|f_{\lambda}\| = 0$ $

My problem now is (as I would like to extend my work), find $ \lambda \in (0,\infty)$ for which $ \|f_{\lambda}\|$ is maximum. Appreciate your suggestions to solve this problem.


The minimization problem from the linked paper is given below for the self containment of the post. If given that $ k>\frac{m}{2}$ , the paper proves that there is a unique minimizer in the set $ S$ .

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It is given that $ k>\frac{m}{2}$