In the Heisenberg group (endowed with its Carnot-Carathéodory structure), we prove that a compact set which satisfies an analog of Peter Jones' geometric lemma is contained in a rectifiable curve. This quantitative condition is given in terms of Heisenberg numbers which measure how well the set is approximated by Heisenberg straight lines.
Cite this article
Fausto Ferrari, Bruno Franchi, Hervé Pajot, The Geometric Traveling Salesman Problem in the Heisenberg Group. Rev. Mat. Iberoam. 23 (2007), no. 2, pp. 437–480