The Art Gallery Guardian

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 \(f:\R^2\to \R^2\), \[ f(x) = (x_1-x_2,x_1+x_2) \] 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 2012-08-23.
Tags: math.