Let $\mathcal{X}$ be a topological space, and let $\tilde{d}$ be a `metric-like' discrepancy, e.g.
- $\tilde{d} :\mathcal{X} \times \mathcal{X} \to [0, \infty]$,
- $d(x, x) = 0$ for all $x$
- $d(x, y) > 0$ for $y \neq x$
- $d(x, y) = d(y, x)$ for all $x, y$
and currently not stipulating the triangle inequality. (I can't remember if this is a semi-metric, or a pseudo-metric, etc.)
Suppose I now want to use $\tilde{d}$ to construct an `actual' metric, by defining
$$d(x, y) = \inf \left\{ \sum_{i = 1}^N \tilde{d} (x_{i-1}, x_i) : N \in \mathbb{N},x_0 = x, x_N = y, x_i \in \mathcal{X} \text{ for } 1 \leqslant i < N \right\}.$$
My question is: does this construction (or something qualitatively similar to it) have a name? Something to do with geodesics, perhaps?
A related question is whether additional constraints on $\tilde{d}$ are needed to ensure that this construction gives a nontrivial metric (i.e. that the infimum in the definition is positive for $y \neq x$); an answer to this would be interesting to me, but not crucial.