Sharp estimate on the inner distance in planar domains

Danka Lučić Department of Mathematics and Statistics, University of Jyvaskyla, Finland Enrico Pasqualetto Department of Mathematics and Statistics, University of Jyvaskyla, Finland Tapio Rajala Department of Mathematics and Statistics, University of Jyvaskyla, Finland

TBD mathscidoc:2203.43012

Arkiv for Matematik, 58, (1), 133-159, 2020.4
We show that the inner distance inside a bounded planar domain is at most the one-dimensional Hausdorff measure of the boundary of the domain. We prove this sharp result by establishing an improved Painlevé length estimate for connected sets and by using the metric removability of totally disconnected sets, proven by Kalmykov, Kovalev, and Rajala. We also give a totally disconnected example showing that for general sets the Painlevé length bound ϰ(E)≤πH^1(E)is sharp.
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@inproceedings{danka2020sharp,
  title={Sharp estimate on the inner distance in planar domains},
  author={Danka Lučić, Enrico Pasqualetto, and Tapio Rajala},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220310170753814438937},
  booktitle={Arkiv for Matematik},
  volume={58},
  number={1},
  pages={133-159},
  year={2020},
}
Danka Lučić, Enrico Pasqualetto, and Tapio Rajala. Sharp estimate on the inner distance in planar domains. 2020. Vol. 58. In Arkiv for Matematik. pp.133-159. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220310170753814438937.
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