Metrics ρ, quasimetrics ρ and pseudometrics inf ρs

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Let ρ be a metric on a space X and let s≥1. The function ρs(a, b) = ρ(a, b)s is a quasimetric (it need not satisfy the triangle inequality). The function inf ρss(a, b) defined by the condition inf ρs(a, b) = inf(σn 0ρs(zi, zi+1) z0 = a, zn = b) is a pseudometric (i.e., satisfies the triangle inequality but can be degenerate). We show how this degeneracy can be connected with the Hausdorff dimension of the space (X,ρ). We also give some examples showing how the topology of the space (X, infρs) can change as s changes.

Original languageEnglish
Pages (from-to)264-272
Number of pages9
JournalConformal Geometry and Dynamics
Issue number10
Publication statusPublished - 1 Jan 2017


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