The traveling salesman problem in the Heisenberg group: Upper bounding curvature
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- by Sean Li and Raanan Schul PDF
- Trans. Amer. Math. Soc. 368 (2016), 4585-4620 Request permission
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.References
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Additional Information
- Sean Li
- Affiliation: Department of Mathematics, The University of Chicago, Chicago, Illinois 60637
- MR Author ID: 899540
- 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
- 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
- © Copyright 2015 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 368 (2016), 4585-4620
- MSC (2010): Primary 28A75, 53C17
- DOI: https://doi.org/10.1090/tran/6501
- MathSciNet review: 3456155