The Traveling Salesman Theorem for Jordan Curves in Hilbert Space

Abstract

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 $\mathbb{R}^n$.