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arXiv · 1506.08888

On a discrete version of length metrics

Pedro Zühlke
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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.

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Pedro Zühlke. 2015-06-29. On a discrete version of length metrics. https://doi.org/10.1007/s40863-016-0039-3

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