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