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

• DOI: 10.1090/tran/6501
• Publication Date: 2015-10-28T13:19:16Z
• Authors: Sean Li; Raanan Schul
• Keywords: Heisenberg group; Lemma (botany); Carnot cycle

We show that if a subset $K$ in the Heisenberg group (endowed with Carnot-Carathéodory metric) is contained rectifiable curve, then it satisfies modified analogue of Peter Jones's geometric lemma. This quantitative version statement finite length curve has tangent at almost every point. condition complements work by Ferrari, Franchi, and Pajot (2007) except power 2 changed to 4. Two key tools we use proof are martingale argument like Schul as well new curvature inequality group.