I need to derive the equation of the ball $B \in M$, where $M = (\mathbb{R}^2, d)$ and $d$ is a Chebyshev distance, i.e. $d(x, y) = \max\left\{|x_1 - y_1|, |x_2 - y_2|\right\}$.
To do so, I need to find the appropriate way of rewriting the $\max$ to some kind of equation/function. What are some most convenient ways of doing so, in algebraic (high school way) or single variable calculus preferably?
You could use
$$\max(a,b) = \frac{a + b + |a - b|}{2},$$
where $a = |x_1 - y_1|$ and $b = |x_2 - y_2|$.