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Is there any example of a metric space $X$ with more than two points such that the triangle inequality is always equality?

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In any metric space with at least two points, the triangle inequality is an actual inequality. For if $x\neq y$ and thus $d(x,y) > 0$, then by the triangle inequality, $$ 0 = d(x,x) \leq d(x,y) + d(y,x) = 2d(x,y), $$ and by our hypothesis we must therefore conclude that $$ d(x,x) < d(x,y) + d(y,x). $$

Edit: removed some redundancies. The same conclusions hold for a nontrivial pseudometric.

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Such an example does not exist ! Suppose that the metric space $X$ contains 2 points $x,y$ with $x \ne y.$

Then we have

$0=d(x,x) < d(x,y)+d(y,x)=2d(x,y)$

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Perhaps no: $0=d(x,x)=d(x,y)+d(y,x)$; so $d(x,y)=0$ i.e. $x=y$.

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