analysis - Showing triangle inequality for a norm

I want to determine whether the following is a norm or not:

$$\begin{equation} \max\{|x_1-x_2|, |x_1+x_2|, |x_3|, |x_4|, \ldots,|x_n|\} \end{equation}$$
Specifically, I want to know whether the triangle inequality holds:
$$\begin{equation} ||x+y|| \leq ||x||+||y|| \end{equation}$$ I also noted that this is very similar to the $L-\infty$ norm:
$$\begin{equation} ||x||_\infty = \max_{1\leq i\leq n}|x_i| \end{equation}$$
Since this norm is identical to the $L-\infty$ norm except for the first two elements, it suffices to consider the cases when $|x_1+x_2|$ and $|x_1-x_2|$ are the maximum

elements.
I am stuck on this thought, not being able to cover all the cases. (I have been thinking what if $x_1$ and $x_2$ are large but $y_1$ and $y_2$ are small. Anyway, this
leads me to believe that there is a simpler way.

1. How do I determine whether the triangle inequality holds?


2. What is a good strategy in general when dealing with maxima?

Answer

Consider the map $f:(x_1,x_2,x_3,\ldots,x_n)\mapsto(x_1-x_2,x_1+x_2,x_3,\ldots,x_n)$. It is easy to show that this is linear and invertible. So, you have got a linear bijection on your hands, and your candidate norm is $|x| = |f(x)|_\infty$.

Then the problem reduces to "given a linear bijection $f:\mathbb{R}^n\to\mathbb{R}^n$ and a known norm $|\cdot|$, show that $|f(\cdot)|$ is also a norm" which (I think) is true and much easier to tackle.