Arkiv för Matematik

Volume 58 (2020)

Number 1

Sharp estimate on the inner distance in planar domains

Pages: 133 – 159

DOI: https://dx.doi.org/10.4310/ARKIV.2020.v58.n1.a9

Authors

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)

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.

Keywords

inner distance, Painlevé length, accessible points

2010 Mathematics Subject Classification

Primary 28A75. Secondary 31A15.

Full Text (PDF format)

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

Received 23 May 2019

Received revised 4 December 2019

Accepted 16 December 2019

Published 21 July 2022