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      Subsets of rectifiable curves in Banach spaces: sharp exponents in Schul-type theorems

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          Abstract

          The Analyst's Traveling Salesman Problem is to find a characterization of subsets of rectifiable curves in a metric space. This problem was introduced and solved in the plane by Jones in 1990 and subsequently solved in higher-dimensional Euclidean spaces by Okikiolu in 1992 and in the infinite-dimensional Hilbert space 2 by Schul in 2007. In this paper, we establish sharp extensions of Schul's necessary and sufficient conditions for a bounded set Ep to be contained in a rectifiable curve from p=2 to 1<p<. While the necessary and sufficient conditions coincide when p=2, we demonstrate that there is a strict gap between the necessary condition and sufficient condition when p2. This investigation is partly motivated by recent work of Edelen, Naber, and Valtorta on Reifenberg-type theorems in Banach spaces and complements work of Hahlomaa and recent work of David and Schul on the Analyst's TSP in general metric spaces.

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          Author and article information

          Journal
          26 February 2020
          Article
          2002.11878
          022172d3-45dc-432a-93ae-2e5e798217b8

          http://arxiv.org/licenses/nonexclusive-distrib/1.0/

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          Custom metadata
          Primary 28A75, Secondary 26A16, 28A80, 46B20, 65D10
          49 pages, 4 figures
          math.CA math.FA math.MG

          Functional analysis,Geometry & Topology,Mathematics
          Functional analysis, Geometry & Topology, Mathematics

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