A relation between $L_\infty$ metric and $L_1$ metric in $\R^2$

While solving the Meeting Point problem from hackerrank, I have stumbled upon a relation between $L_\infty$ metric and $L_1$ metric in $\R^2$.

Theorem1

Let $x=(x_1,x_2)$, consider the map $f:\R^2\to \R^2$, $f(x) = (x_1-x_2,x_1+x_2)$ then $d_\infty(x,y) = \frac{d_1(f(x),f(y))}{2}$

Proof
1. Notice we only need to prove $\|x\|_\infty = \frac{ \|f(x)\|_1}{2}$ as the metric is the standard metric generated by the norm.
2. Note the following relation $\max(|a|,|b|) = \frac{ |a+b| + |a-b|}{2}$.
3. Combine the two above and we get the result.
Posted by Chao Xu on .
Tags: math.