On a discrete version of length metrics

Pedro Zühlke

doi:10.1007/s40863-016-0039-3

paper

On a discrete version of length metrics

arXiv:1506.08888 · doi:10.1007/s40863-016-0039-3

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.

arXiv comment9 pages, 1 figure