# 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.

• I suggest you try finding an example where $X$ has 3 points. – Eric Wofsey Nov 15 '15 at 6:35
• what properties of a metric does the minimum have? Wha properies are you not able to prove? – miracle173 Nov 15 '15 at 6:46

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