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 $\mathbb{U}$ of a metric space $X$, the predecessor $\mathbb{U}_\ast$ of $\mathbb{U}$ is defined by doubling the radii of all open balls contained inside $\mathbb{U}$, and taking their union. The predecessor of $\mathbb{U}$ is an open set containing $\mathbb{U}$. The directed doubling distance between $\mathbb{U}$ and another subset $\mathbb{V}$ is the number of times that the predecessor operation needs to be applied to $\mathbb{U}$ to obtain a set that contains $\mathbb{V}$. Finally, the doubling distance between open sets $\mathbb{U}$ and $\mathbb{V}$ is the maximum of the directed distance between $\mathbb{U}$ and $\mathbb{V}$ and the directed distance between $\mathbb{V}$ and $\mathbb{U}$.
Funding Statement
V.S. has been partly supported by the Academy of Finland via the Centre of Excellence in Analysis and Dynamics Research.
Citation
János Flesch. Arkadi Predtetchinski. Ville Suomala. "The doubling metric and doubling measures." Ark. Mat. 58 (2) 243 - 266, October 2020. https://doi.org/10.4310/ARKIV.2020.v58.n2.a2
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