I have two functions {d1,d2}, where d1 satisfies all metric properties, but one: triangle inequality. However, the following holds:
d1(a,b) ≤ d2(a,c) + d1(b,c)
Can I say d1 is a kind of metric? Is there a formal definition for it?
I know a super metric (or ultra metric) is an example of a function that satisfies a stronger constraint of the triangle inequality, where d(a, b) ≤ max{d(a, c),d( b, c)}. So maybe there is also a formal definition for the case involving d1.