Is any norm on $\mathbb R^n$ invariant with respect to componentwise absolute value?

Question

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.

Answer 1

You may consider, for example,

$$\|(x_1,x_2)\|:=|x_1|+|x_1-x_2|.$$