Metrics ρ, quasimetrics ρs and PSEUDOMETRICS inf ρs

Abstract. 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. © 2017 American Mathematical Society.