# A relation between $L_{∞}$ metric and $L_{1}$ metric in $R_{2}$

While solving the Meeting Point problem from hackerrank, I have stumbled upon a relation between $L_{∞}$ metric and $L_{1}$ metric in $R_{2}$.

Theorem1

Let $x=(x_{1},x_{2})$, consider the map $f:R_{2}→R_{2}$, $f(x)=(x_{1}−x_{2},x_{1}+x_{2})$ then $d_{∞}(x,y)=2d_{1}(f(x),f(y)) $

Proof

- Notice we only need to prove $∥x∥_{∞}=2∥f(x)∥_{1} $ as the metric is the standard metric generated by the norm.
- Note the following relation $max(∣a∣,∣b∣)=2∣a+b∣+∣a−b∣ $.
- Combine the two above and we get the result.