I want to minimize the difference between two norms, for $h \in \mathbb{R}^n$ and $M, N \in \mathbb{R}^{m \times n}$. $||\cdot||$ is the $L_2$ norm on $\mathbb{R}^m$: $||x|| = \sqrt{ \displaystyle\sum_{i=1}^m x_i^2 } $
$ h^* = \displaystyle\arg\min_{||h|| = 1} \Big\{ ||Mh|| - ||Nh|| \Big\}$
Is there a way to find an expression for $h^*$ depending on $M$ and $N$ ?
All the references that I have found deal with, either the difference in the norm, like $||N - Mh||$ or the weights of the norms are positive like $||Mh|| + w||Nh|| \quad \text{with} \; w > 0$. So they are not relevant, unless there is a way to transform my problem in something more classic ?