6533b7cffe1ef96bd1258f8c

RESEARCH PRODUCT

Sharp estimate on the inner distance in planar domains

Tapio RajalaDanka LučićEnrico Pasqualetto

subject

Pure mathematicsMathematics - Complex VariablesGeneral MathematicsBoundary (topology)accessible pointsMetric Geometry (math.MG)31A15Domain (mathematical analysis)inner distancePlanarMathematics - Metric GeometryPrimary 28A75. Secondary 31A15Bounded functionTotally disconnected spaceMetric (mathematics)FOS: Mathematics28A75Hausdorff measureComplex Variables (math.CV)Painlevé lengthMathematics

description

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 Painlev\'e 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 Painlev\'e length bound $\kappa(E) \le\pi \mathcal{H}^1(E)$ is sharp.

yearjournalcountryeditionlanguage
2020-04-01
https://dx.doi.org/10.48550/arxiv.1905.07988