Subsets of rectifiable curves in Banach spaces I: sharp exponents in traveling salesman theorems

Abstract

The Analyst's Traveling Salesman Problem is to find a characterization of subsets of rectifiable curves in a metric space. This problem was introduced and solved in the plane by Jones in 1990 and subsequently solved in higher-dimensional Euclidean spaces by Okikiolu in 1992 and in the infinite-dimensional Hilbert space ${\ell }_2$ by Schul in 2007. In this paper, we establish sharp extensions of Schul's necessary and sufficient conditions for a bounded set $E\subset {\ell }_p$ to be contained in a rectifiable curve from $p=2$ to $1<p<\infty$. While the necessary and sufficient conditions coincide when $p=2$, we demonstrate that there is a strict gap between the necessary condition and sufficient condition when $p\ne 2$. We also identify and correct technical errors in the proof by Schul. This investigation is partly motivated by recent work of Edelen, Naber, and Valtorta on Reifenberg-type theorems in Banach spaces and complements work of Hahlomaa and recent work of David and Schul on the Analyst's TSP in general metric spaces.

66 pages, 4 figures, v3: modified definition of *-almost-flat arcs and proof of Lemma 3.29; split the main theorem into Theorem 1.6 (sufficient conditions) and Theorem 1.7 (necessary conditions); we prove sufficient conditions in full and reduce necessary conditions to Theorem 3.30, which we prove in Part 2, see arXiv:2208.10288; moved portion of Appendix A to Part 2