Subsets of Rectifiable curves in Hilbert Space-The Analyst's TSP

Abstract

We study one dimensional sets (Hausdorff dimension) lying in a Hilbert space. The aim is to classify subsets of Hilbert spaces that are contained in a connected set of finite Hausdorff length. We do so by extending and improving results of Peter Jones and Kate Okikiolu for sets in $\R^d$. Their results formed the basis of quantitative rectifiability in $\R^d$. We prove a quantitative version of the following statement: a connected set of finite Hausdorff length (or a subset of one), is characterized by the fact that inside balls at most scales around most points of the set, the set lies close to a straight line segment (which depends on the ball). This is done via a quantity, similar to the one introduced in \cite{Jones90}, which is a geometric analog of the Square function. This allows us to conclude that for a given set $K$, the $\ell_2$ norm of this quantity (which is a function of $K$) has size comparable to a shortest (Hausdorff length) connected set containing $K$. In particular, our results imply that, with a correct reformulation of the theorems, the estimates in \cite{Jones90,Okik92} are {\bf independent of the ambient dimension}.