The traveling salesman problem in the Heisenberg group: upper bounding curvature. (English) Zbl 1350.53044
Summary: 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 F. Ferrari et al. [Rev. Mat. Iberoam. 23, No. 2, 437–480 (2007; Zbl 1142.28004)] 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 the second author [J. Anal. Math. 103, 331–375 (2007; Zbl 1152.28006)] as well as a new curvature inequality in the Heisenberg group.
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