Hello everyone.

I am studying a bivariate function $d$ on a set $\mathcal{X}\times \mathcal{X}$ which is **symmetric** and satisfies all **triangle inequalities** but does not agree with the coincidence axiom (which says that $d(x,y)=0\Leftrightarrow x=y$).

Pseudodistances are such that only $d(x,y)=0 \Leftarrow x=y$.

In the case I am concerned with,
the function is actually such that $d(x,y)>0$ for **all** $x,y$ (including $d(x,x)>0$) and so only the implication $\Rightarrow$ would be valid, trivially because $d(x,y)$ is never 0.

I have been looking for a proper generalization of distance functions that would take into account that case but I have not found any. Do such functions bear a name and have they ever been considered in the context of generalized metrics?

Many thanks in advance, Marco

psychodistance;) – Pietro Majer Apr 28 '13 at 11:01