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

Li, Sean; Schul, Raanan

doi:10.1090/tran/6501

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.

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