[Math] Is the minimum of two metrics is again a metric

Question

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.

Answer 1

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 $d(a,b)+d(b,c)< d(a,c)$.