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

@article{Li2013TheTS,
  title={The traveling salesman problem in the Heisenberg group: upper bounding curvature},
  author={Sean Li and Raanan Schul},
  journal={arXiv: Metric Geometry},
  year={2013}
}
We show that if a subset $K$ in the Heisenberg group (endowed with the Carnot-Carath\'{e}odory 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 \cite{FFP} 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… 

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