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For a metric space with distance $d$, for any 3 elements $x_1,x_2,x_3$, there exists a permutation $\pi$ such that $d(x_{\pi(1)},x_{\pi(3)})=d(x_{\pi(1)},x_{\pi(2)})+d(x_{\pi(2)},x_{\pi(3)})$.

Is there a name for such metric space?

One such metric would be points on the real line, but there are probably more general metrics.

Chao Xu
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  • Any subset of the real line will work too. I do not think there is anything else. – Moishe Kohan Aug 04 '23 at 15:57
  • I originally thought all such metric spaces can be embedded into a real line, but then I found a counter-example: let $X={(0,0),(0,1),(1,0),(1,1)}$ and define $$d(x,y)=\begin{cases}1&\text{if $x$ and $y$ lie on the same vertical side}\\frac{1}{2}&\text{else if $x\neq y$}\0&\text{if $x=y$}\end{cases}$$ It is easy to check this gives rise to a metric space. Since any choice of 3 distinct points of $X$ contains two points - say $x$ and $z$ - on the same vertical side and the remaining point - say $y$ - on the opposite side, we get $d(x,z)=1=d(x,y)+d(y,z)$. – Sangchul Lee Aug 04 '23 at 16:04

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