Metric induced by infinity norm? The infinity norm is defined as follows:
$||\mathbf{x}||=\max\{|x_1|,...,|x_n|\}$.
Say we have $\mathbb{R}^2$ endowed with $||\ ||_{\infty}$. Then what is $d((2,1),(3,2))$? Is it just $(3,2)$? If so, then how does it make sense to talk about a ball in this metric?

A metric is always a mapping from some space to $\mathbb R$, so your answer $(3,2)$ cannot be correct. The distance must always be a single number, not a pair of numbers.
That said the answer $d((2,1), (3,2)) = 1$. Why? Well,

If $\|.\|$ is a norm, then the metric induced by it is defined as $$d(x,y) = \|x-y\|$$

Now, in your case $x=(2,1), y=(3,2)$. The distance between them is $d(x,y) = \|x-y\|$ which is easy to calculate since $x-y$ is easy to calculate.
