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The traveling salesman problem in the Heisenberg group: Upper bounding curvature


Authors: Sean Li and Raanan Schul
Journal: Trans. Amer. Math. Soc. 368 (2016), 4585-4620
MSC (2010): Primary 28A75, 53C17
DOI: https://doi.org/10.1090/tran/6501
Published electronically: October 28, 2015
MathSciNet review: 3456155
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Abstract: We show that if a subset $ K$ in the Heisenberg group (endowed with the Carnot-Carathéodory metric) is contained in a rectifiable curve, then it satisfies a modified analogue of Peter Jones's geometric lemma. This is a quantitative version of the statement that a finite length curve has a tangent at almost every point. This condition complements that of a work by Ferrari, Franchi, and Pajot (2007) except a power 2 is changed to a power 4. Two key tools that we use in the proof are a geometric martingale argument like that of Schul (2007) as well as a new curvature inequality in the Heisenberg group.


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Additional Information

Sean Li
Affiliation: Department of Mathematics, The University of Chicago, Chicago, Illinois 60637
Email: seanli@math.uchicago.edu

Raanan Schul
Affiliation: Department of Mathematics, Stony Brook University, Stony Brook, New York 11794-3651
Email: schul@math.sunysb.edu

DOI: https://doi.org/10.1090/tran/6501
Keywords: Heisenberg group, traveling salesman theorem, Jones $\beta$ numbers, curvature
Received by editor(s): June 28, 2013
Received by editor(s) in revised form: January 15, 2014, and May 9, 2014
Published electronically: October 28, 2015
Article copyright: © Copyright 2015 American Mathematical Society