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Let $X$ be $\mathbb{R}^3$ with the sup norm $\|\cdot\|_{\infty}$. Let $Y=\{x\in X: \|x\|_{\infty}=1\}$. For $x,y\in Y,y\neq -x$ define $d(x,y)$ to be the arc length of the path $$Y\cap \{\lambda x+\mu y: \lambda\ge 0, \mu\ge 0\}.$$ Define $d(x,-x)=4$. Note that the arc lengths are computed using the sup norm. My question is: Does $d$ define a metric on $Y$? A related question was answered in Shortest path on unit sphere under $\|\cdot\|_\infty$

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    In fact the answer can be found in the given link. Let $A=(1,3/4,1/4), B=(3/4,1,3/4), C=(1,1,1/2)$ and $M=(1,1,4/7)$. Then $d(A,C)=1/4, d(C,B)=1/4$ and $d(A,B)=d(A,M)+d(M,B)=4/7$. So $d(A,C)+d(C,B), proving that $d$ is not a metric.2012-02-11
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    TCL: Why not post an answer?2012-02-11
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    @TCL I'm not sure what the relevance is more than a year later, but it's better to ask a new question instead of editing in this case -- your edit made this question wholly different from its original.2013-11-11

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