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.

中文翻译：

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对平面域内距离的清晰估计

我们表明，有界平面域内部的内部距离最多是域边界的一维Hausdorff度量。我们通过为连接集建立改进的Painlevé长度估计并使用完全断开集的度量可移动性来证明这一尖锐的结果，这由Kalmykov，Kovalev和Rajala证明。我们还给出了一个完全脱节的示例，该示例显示对于一般集合，Painlevé长度限制$ \ varkappa（E）\ leq \ pi \ mathcal {H} ^ 1（E）$是尖锐的。