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A relation between LL_\infty metric and L1L_1 metric in R2\R^2


While solving the Meeting Point problem from interviewstreet, I have stumbled upon a relation between LL_\infty metric and L1L_1 metric in R2\R^2.

Theorem1

Let x=(x1,x2)x=(x_1,x_2), consider the map f:R2R2f:\R^2\to \R^2, f(x)=(x1x2,x1+x2)\displaystyle f(x) = (x_1-x_2,x_1+x_2) then d(x,y)=d1(f(x),f(y))2\displaystyle d_\infty(x,y) = \frac{d_1(f(x),f(y))}{2}

Proof
  1. Notice we only need to prove x=f(x)12\|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)=a+b+ab2\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.