The Traveling Salesman Theorem for Jordan Curves in Hilbert Space

TL;DR

This paper proves a version of the Traveling Salesman Theorem in Hilbert space, establishing a quantitative relationship between the shortest curve length containing a set and its flatness, with improvements for Jordan arcs.

Contribution

It provides a complete proof of Schul's necessary condition in Hilbert space and refines the theorem for Jordan arcs, extending and sharpening prior results.

Findings

Full proof of Schul's necessary half in Hilbert space

Sharpened quantitative relationship for Jordan arcs

Extension of classical TSP results to infinite-dimensional spaces

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 $\mathbb{R}^n$.