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

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-regularwhere β 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 ⊆ 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. Contents 1. Introduction 1 2. Preliminaries 5 3. Reifenberg parametrisations in Hilbert space 12 4. Estimating curvature for Reifenberg flat sets 16 5. Estimating curvature for general sets 23 6. Estimating measure 32 7. Bilateral Weak Geometric Lemma 38 Appendix A. Parameterisation in Hilbert space 43 Appendix B. Constants 54 Appendix C. Reduction to Euclidean space 56 References 61

Rectifiability; a survey

“…In [AS18] Azzam and Schul obtained higher dimensional analogues of both theorems 4.16 and 4.17 using β-integrals defined in terms of Hausdorff content. Hilbert space versions were proven by Hyde in [Hyd21]. where the infima are taken over all m-planes in R n .…”