A $d$-dimensional Analyst's Travelling Salesman Theorem for subsets of Hilbert space

TL;DR

This paper extends the Analyst's Travelling Salesman Theorem to infinite dimensional Hilbert spaces, providing a quantitative framework for rectifiability and curvature of sets in this setting.

Contribution

It adapts the Reifenberg Parametrization Theorem for Hilbert spaces and establishes a characterization of uniform rectifiability via the Bilateral Weak Geometric Lemma.

Findings

Proves a Hilbert space version of the Analyst's Travelling Salesman Theorem.

Shows equivalence between uniform rectifiability and the Bilateral Weak Geometric Lemma in Hilbert spaces.

Provides a modified Reifenberg parametrization applicable in infinite dimensions.

Abstract

We are interested in quantitative rectifiability results for subsets of infinite dimensional Hilbert space $H$. We prove a version of Azzam and Schul's $d$-dimensional Analyst's Travelling Salesman Theorem in this setting by showing for any lower $d$-regular set $E \subseteq H$ that \[ \text{diam}(E)^d + \beta^d(E) \sim \mathscr{H}^d(E) + \text{Error}, \] where $\beta^d(E)$ give a measure of the curvature of $E$ and the error term is related to the theory of uniform rectifiability (a quantitative version of rectifiability introduced by David and Semmes). To do this, we show how to modify the Reifenberg Parametrization Theorem of David and Toro so that it holds in Hilbert space. As a corollary, we show that a set $E \subseteq H$ is uniformly rectifiable if and only if it satisfies the so-called Bilateral Weak Geometric Lemma, meaning that $E$ is bi-laterally well approximated by planes at most scales and locations.