The Traveling Salesman Theorem for Jordan Curves in Hilbert Space

• Published in: arXiv.org 2022-10
• Main Author: Krandel, Jared
• Format: Article
• Language: English
• Subjects: Graphs; Hilbert space; Sharpening; Theorems
• ISSN: 2331-8422

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 \(\mathbb{R}^n\).