The Traveling Salesman Theorem for Jordan Curves in Hilbert Space

Given a metric space \(X\), an Analyst's Traveling Salesman Theorem for \(X\) gives a quantitative relationship between the length of a shortest curve containing any subset \(E\subseteq X\) and a multi-scale sum measuring the ``flatness'' of \(E\). The first such theorem was proven by...

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Bibliographic Details
Published in:arXiv.org 2022-10
Main Author: Krandel, Jared
Format: Article
Language:English
Subjects:
Graphs
Hilbert space
Sharpening
Theorems
Online Access:Get full text
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Summary:Given a metric space \(X\), an Analyst's Traveling Salesman Theorem for \(X\) gives a quantitative relationship between the length of a shortest curve containing any subset \(E\subseteq X\) and a multi-scale sum measuring the ``flatness'' of \(E\). The first such theorem was proven by Jones for \(X = \mathbb{R}^2\) and extended to \(X = \mathbb{R}^n\) by Okikiolu, while an analogous theorem was proven for Hilbert space, \(X = H\), by Schul. Bishop has since shown that if one considers Jordan arcs, then the quantitative relationship given by Jones' and Okikioulu's results can be sharpened. This paper gives a full proof of Schul's original necessary half of the traveling salesman theorem in Hilbert space and provides a sharpening of the theorem's quantitative relationship when restricted to Jordan arcs analogous to Bishop's aforementioned sharpening in \(\mathbb{R}^n\).
ISSN:2331-8422