Is any norm on $\mathbb R^n$ invariant with respect to componentwise absolute value? Given $\mathbf{x}=(x_1,...,x_n) \in \mathbb{R}^n$ , define  $ \mathbf{x}'=(|x_1|,...,|x_n|) $ .
Then, is it $||\mathbf{x}'|| = ||\mathbf{x}||$  for every norm on $ \mathbb{R}^n $  ?
NB: The answer is trivially yes for $p$-norms.

You may consider, for example,
$$\|(x_1,x_2)\|:=|x_1|+|x_1-x_2|.$$
