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