Is norm non-decreasing in each variable?

We know that $\|x\|_1=|x_1|+|x_2|$ is norm on $\mathbb R^2$. (It is called $\ell_1$-norm or taxicab norm.) It is easy to see that rotation does not change properties of norm.

So for any angle $\varphi$ the function $$\|x\|=|x_1\cos\varphi-x_2\sin\varphi|+|x_1\sin\varphi+x_2\cos\varphi|$$ is a norm on $\mathbb R^2$.

For $\varphi=\frac\pi 6$ we have $$\|x\|=\frac{|\sqrt3x_1-x_2|+|x_1+\sqrt3x_2|}2.$$

If we fix $x_2=1$, then this function is not monotone in $x_1$, as we can check by plotting |sqrt(3)t-1|+|t+sqrt(3)| in WA.