Revista Matemática Iberoamericana


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Volume 32, Issue 2, 2016, pp. 391–417
DOI: 10.4171/RMI/889

Published online: 2016-06-08

An upper bound for the length of a traveling salesman path in the Heisenberg group

Sean Li[1] and Raanan Schul[2]

(1) University of Chicago, USA
(2) Stony Brook University, USA

We show that a sufficient condition for a subset $E$ in the Heisenberg group (endowed with the Carnot–Carathéodory metric) to be contained in a rectifiable curve is that it satisfies a modified analogue of Peter Jones’s geometric lemma. Our estimates improve on those of [6], allowing for any power $r$ < 4 to replace the power 2 of the Jones-$\beta$-number. This complements in an open ended way our work in [13], where we showed that such an estimate was necessary, however with the power $r$ = 4.

Keywords: Heisenberg group, traveling salesman theorem, Jones $\beta$ numbers, curvature

Li Sean, Schul Raanan: An upper bound for the length of a traveling salesman path in the Heisenberg group. Rev. Mat. Iberoam. 32 (2016), 391-417. doi: 10.4171/RMI/889