Q:

Is the minimum of two metrics is again a metric? 

Let $d_1$ and $d_2$ be two metrics on non empty set $X$.
Is $d$ = $\min\{d_1, d_2\}$ is again metric on $X$? 
I'm looking for a counter example with minimum of two metrics  not being a metric. 

A:

Consider two metrics on the set $\{a,b,c\}$ that have the same distance $d_i(a,c)$ and satisfy $d_i(a,b)+d_i(b,c)=d_i(a,c)$. If these metrics are not identical, their minimum will fail the triangle inequality, with $d(a,b)+d(b,c)< d(a,c)$. 

