Subsets of Rectifiable curves in Hilbert Space-The Analyst’s TSP
نویسنده
چکیده
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 Rd . Their results formed the basis of quantitative rectifiability in Rd . 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 [Jon90], which is a geometric analog of the Square function. This allows us to conclude that for a given set K, the l2 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 [Jon90, Oki92] are independent of the ambient dimension.
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