Inserted: 21 may 2019
Last Updated: 21 may 2019
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.
Keywords: Inner distance, Painlevé length, accessible points