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April 2020 Sharp estimate on the inner distance in planar domains
Danka Lučić, Enrico Pasqualetto, Tapio Rajala
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Ark. Mat. 58(1): 133-159 (April 2020). DOI: 10.4310/ARKIV.2020.v58.n1.a9


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 $\varkappa (E) \leq \pi \mathcal{H}^1 (E)$ is sharp.

Funding Statement

All authors partially supported by the Academy of Finland, projects 274372, 307333, 312488, and 314789.


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Danka Lučić. Enrico Pasqualetto. Tapio Rajala. "Sharp estimate on the inner distance in planar domains." Ark. Mat. 58 (1) 133 - 159, April 2020.


Received: 23 May 2019; Revised: 4 December 2019; Published: April 2020
First available in Project Euclid: 16 January 2021

Digital Object Identifier: 10.4310/ARKIV.2020.v58.n1.a9

Primary: 28A75
Secondary: 31A15

Rights: Copyright © 2020 Institut Mittag-Leffler


Vol.58 • No. 1 • April 2020
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