Sharp estimate on the inner distance in planar domains

created by pasqualetto on 21 May 2019

[BibTeX]

Preprint

Inserted: 21 may 2019
Last Updated: 21 may 2019

Year: 2019

ArXiv: 1905.07988 PDF

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 $\kappa(E)\leq\pi\mathcal H^1(E)$ is sharp.