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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

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What will you do with this distance? Will you use it to define a topology (how?), or maybe you want to modify it so as to get a true distance out of it? In any case, if I think of a point with a positive distance from itself, the name that occurs naturally to my mind for this distance is, in analogy, psychodistance ;) – Pietro Majer Apr 28 '13 at 11:01
Hi Marco: I'd call these distances "strictly positive definite" but that sounds like mixing up terms. – Suvrit Apr 28 '13 at 18:23
One example would be: $d(x,y) := \delta(x,y)+\delta(x,a)+\delta(y,b)$, where $\delta(x,y)$ is a usual metric, and $a \neq b$ are two arbitrary points in space (to have an example slightly less trivial than $d(x,y) := \delta(x,y) + \epsilon$... – Suvrit Apr 28 '13 at 18:36
I never seeing this either, but I can imagine that such things appear in some applied math problems where "distance" is a "penalty" and you penalized not only by moving to another place, but also by "staying put", e.g.: If your car stays in the garage, it still depreciates and you still have to pay taxes. (In order to satisfy triangle inequalities, there will be also a strict lower bound on how far you have to travel once you leave the garage.) Other examples would come from biology or economics; you may have to pay your mutual fund for managing even if you neither buy nor sell. – Misha Apr 28 '13 at 18:37
Thanks for all your comments! I gave it some extra thought. I see two links: negative definite kernels (in the sense of Berg Christensen Ressel. They consider negative definite kernels $\psi$ (p.82) that may not be such that $\psi(x,x)=0$. Negative definite kernels and distances are different, but they are somewhat related (the bigger, the more different). The other thing that's easy to check is that $1_{x\ne y} d(x,y)$ is itself a distance. It's not continuous.. but still a distance! – mcuturi Apr 29 '13 at 4:49

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