diameter of graph with taxicab metric

Question

I was just asked the following amusing question. Consider $\mathbb{R}^2$ with the taxicab metric (the taxicab distance between $(a,b)$ and $(c,d)$ is $|a-c|+|b-d|$). Given a finite set of points in $\mathbb{R}^2$ such that the taxicab distance between any two of them is at most $D$, is there a point in $\mathbb{R}^2$ which has taxicab distance at most $D/2$ from all of them?

Answer 1

Found a fun solution. First, note that this question is equivalent to the same question asked about the $ \mathcal{l}^\infty $ metric on $ \mathbb{R}^2 $ since $ (\mathbb{R}^2, \mathcal{l}^\infty) $ is isometric to $ (\mathbb{R}^2, \mathcal{l}^1) $ (the taxicab space).

In $ (\mathbb{R}^2, \mathcal{l}^\infty ) $ the argument is pretty easy to write down. If $ S $ is your subset of $ \mathbb{R}^2 $, define: $$ a = \inf_{(x,y)\in S}{x}, \ \ b = \sup_{(x,y)\in S}{x},\ \ c = \inf_{(x,y)\in S}{y},\ \ d = \sup_{(x,y)\in S}{y} $$ and take your central point $ p = (\frac{a+b}{2}, \frac{c+d}{2}) $.