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

While solving the Meeting Point problem from interviewstreet, 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, \displaystyle f(x) = (x_1-x_2,x_1+x_2) then \displaystyle 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 2012-08-23.
Tags: math.