Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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 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.

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The categories of this paper are: cond-mat
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The categories of this paper are: hep-ex
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The categories of this paper are: stat
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The categories of this paper are: solv-int
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The categories of this paper are: astro-ph
35
The categories of this paper are: mtrl-th
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The categories of this paper are: q-alg
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The categories of this paper are: patt-sol
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The categories of this paper are: bayes-an
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The categories of this paper are: comp-gas
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The categories of this paper are: nlin
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The categories of this paper are: q-fin
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The categories of this paper are: plasm-ph
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The categories of this paper are: hep-ph
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The categories of this paper are: chao-dyn
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The categories of this paper are: gr-qc