The doubling metric and doubling measures

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Abstract

We introduce the so-called doubling metric on the collection of non-empty bounded open subsets of a metric space. Given an open subset U of a metric space X, the predecessor U∗ of U is defined by doubling the radii of all open balls contained inside U, and taking their union. The predecessor of U is an open set containing U. The directed doubling distance between U and another subset V is the number of times that the predecessor operation needs to be applied to U to obtain a set that contains V. Finally, the doubling distance between open sets U and V is the maximum of the directed distance between U and V and the directed distance between V and U.
Original languageEnglish
Pages (from-to)243-266
JournalArkiv for Matematik
Volume58
Issue number2
DOIs
Publication statusPublished - 2020

Keywords

  • metric
  • doubling measure
  • quasisymmetric map

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