- journal article metadata
European Mathematical Society Publishing House
2016-09-19 17:05:48
Revista Matemática Iberoamericana
Rev. Mat. Iberoamericana
RMI
0213-2230
2235-0616
General
10.4171/RMI
http://www.ems-ph.org/doi/10.4171/RMI
subscribers
European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society (from 2012)
32
2016
2
An upper bound for the length of a traveling salesman path in the Heisenberg group
Sean
Li
University of Chicago, CHICAGO, UNITED STATES
Raanan
Schul
Stony Brook University, STONY BROOK, UNITED STATES
Heisenberg group, traveling salesman theorem, Jones $\beta$ numbers, curvature
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.
Measure and integration
Differential geometry
391
417
10.4171/RMI/889
http://www.ems-ph.org/doi/10.4171/RMI/889