On a discrete version of length metrics

Abstract: Let $ (X,d) $ be a metric space. We study a metric $ d_0 $ on $ X $ naturally derived from $ d $. If $ (X,d) $ is complete and locally compact, or if it is complete and $ (d_0)_0=d_0 $, then $ d_0 $ coincides with the length metric induced by $ d $. Counterexamples are constructed when any of the hypotheses is absent. The behavior of the iterates of $ d_0 $ (the metrics $ d_0^n $ inductively defined as $ (d_0^{n-1})_0 $) is also considered.