An upper bound for the length of a Traveling Salesman path in the Heisenberg group
Abstract
We show that a sufficient condition for a subset E in the Heisenberg group (endowed with the Carnot-Carath\'eodory 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 FFP, by replacing the power 2 of the Jones-β-number with any power r<4. This complements (in an open ended way) our work Li-Schul-beta-leq-length, where we showed that such an estimate was necessary, but with r=4.
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