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In the Heisenberg group H (endowed with its Carnot-Carathéodory structure), we prove that a compact set E ⊂ H which satisfies an analog of Peter Jones’ geometric lemma is contained in a rectifiable curve. This quantitative condition is given in terms of Heisenberg β numbers which measure how well the set E is approximated by Heisenberg straight lines.

The Geometric Traveling Salesman Problem in the Heisenberg Group