We are interested in characterizing the compact sets of the Heisenberg group that are contained in a curve of finite length. Ferrari, Franchi and Pajot recently gave a sufficient condition for those sets, adapting a necessary and sufficient condition due to P. Jones in the Euclidean setting. We prove that this condition is not necessary.
"A counterexample for the geometric traveling salesman problem in the Heisenberg group." Rev. Mat. Iberoamericana 26 (3) 1035 - 1056, September, 2010.