Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The Traveling Salesman Theorem for Jordan Curves in Hilbert Space   Let $\Gamma\subseteq H$ be a Jordan arc in a Hilbert space $H$ with distance
between its endpoints given by $\text{crd}(\Gamma)$ and length given by
$\ell(\Gamma)$. We prove
  \[\sum_{Q\in\hat{\mathcal{G}}}\beta(Q)^2\text{diam}(Q) \simeq \ell(\Gamma) -
\text{crd}(\Gamma)\] where $\hat{\mathcal{G}}$ is a multiresolution family of
balls centered on $\Gamma$. This provides a sharpening of Schul's original
Hilbert space traveling salesman theorem analogous to Bishop's sharpening of
the traveling salesman theorem in $\mathbb{R}^n$ in the case of a Jordan arc.
Specifically, the most significant improvement comes from strengthening Schul's
result in the case of a Jordan arc by replacing $\ell(\Gamma)$ in the
$\lesssim$ direction with $\ell(\Gamma) - \text{crd}(\Gamma)$.

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The categories of this paper are: funct-an
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The categories of this paper are: ao-sci
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The categories of this paper are: q-bio
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The categories of this paper are: quant-ph
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The categories of this paper are: q-alg
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The categories of this paper are: chao-dyn
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The categories of this paper are: alg-geom
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The categories of this paper are: econ
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The categories of this paper are: hep-lat
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The categories of this paper are: patt-sol
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The categories of this paper are: hep-th
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The categories of this paper are: supr-con
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The categories of this paper are: hep-ex
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The categories of this paper are: nlin
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The categories of this paper are: math-ph
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The categories of this paper are: q-fin