平面域内距离的尖锐估计

Pub Date : 2019-05-20 DOI:10.4310/arkiv.2020.v58.n1.a9

Danka Luvci'c, Enrico Pasqualetto, T. Rajala

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引用次数: 1

摘要

我们证明了有界平面区域内的距离最多是该区域边界的一维豪斯多夫测度。我们通过建立一个改进的连通集的Painleve长度估计和使用Kalmykov, Kovalev和Rajala证明的完全不连通集的度量可移除性来证明这个尖锐的结果。我们还给出了一个完全不相关的例子，表明对于一般设置painlevel长度界限$\kappa(E) \le\pi \mathcal{H}^1(E)$是尖锐的。

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Sharp estimate on the inner distance in planar domains

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

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