On a discrete version of length metrics

TL;DR

This paper investigates a naturally derived discrete length metric $d_0$ from a given metric space $(X,d)$, exploring conditions under which $d_0$ coincides with the length metric and analyzing the behavior of its iterates.

Contribution

It introduces a new discrete length metric $d_0$, characterizes when it matches the classical length metric, and examines the properties of its iterative application.

Findings

$d_0$ coincides with the length metric under certain completeness and compactness conditions.

Counterexamples show failure of coincidence when hypotheses are absent.

Iterates of $d_0$ exhibit specific convergence or divergence behaviors.

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.