On a discrete version of length metrics (Q2404591)

In a metric space \((X,d)\) the author considers several notions of induced metric: the length metric \(\bar d\) induced by \(d\), for \(\varepsilon>0\) the metric \(d_\varepsilon\), which is the infimum of the lengths of \(\varepsilon\)-chains between two points, and the metric \(d_0\), which is the supremum of the \(d_\varepsilon\)s. He investigates the relationship between these metrics. One always has that \(d\leq d_\varepsilon\leq d_0\leq \bar d\). It is shown that in a complete metric space \((X, d)\) the metrics \(d_0\) and \(\bar d\) coincide if \((X, d)\) is locally compact or \((d_0)_0 = d_0\). In the former case, \((X, d_0)\) is a geodesic space. The author furthermore provides counterexamples where \(d_0<\bar d\) if one of the above assumptions is not satisfied (i.e., \((X,d)\) is not complete or not locally compact or \(d_0<(d_0)_0\)). In the last section of the paper iterates of \(d_0\) are considered where \(d_0^n\) is recursively defined as \((d^{n-1}_0 )_0\). If the sequence of iterates of \(d_0\) in a complete metric space becomes stationary, then the stationary term equals \(\bar d\). Examples of complete metric spaces are given such that \(d < d_0 < \cdots < d^n_0 < d^{n+1}_0= \bar d\) and such that \(\lim_n d^n_0 <\bar d\).