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

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.