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suppose $X$ is a set and $e$, $f$ are two metric on the set $X$. Then I knew that $$(x,y)\rightarrow \min\{e(x,y),f(x,y)\}$$ is not a metric on $X$. There are counterexamples in which triangle inequality fails. But I want to know under what condition this will be a metric on $X$.

Thanks for help in advance.

Przemysław Scherwentke
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Yogi
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    Note that the two other axioms hold automatically, so it's just a matter of determining the condition under which $\min{e, f}$ satisfies the triangle inequality when $e, f$ are metrics. – Travis Willse Jul 19 '15 at 05:46

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