Subsets of Rectifiable curves in Hilbert Space and the Analyst's TSP

Raanan Schul
Yale University

We characterize subsets of Hilbert space that are contained in a curve of finite 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 show that, given a set K , we have diam(K) + Sum of ß^2(Q)diam(Q) ~ Length(Gamma MST). Here ß (the Jones Beta Number) is taken with respect to K, the sum is over a multiresolution family of (overlapping) balls Q centered on K, and Gamma MST is the shortest connected set containing K.

Presentation (PDF File)

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