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The geometric traveling salesman problem in the Heisenberg group. (English) Zbl 1142.28004
Let \(H\) be the first Heisenberg group endowed with its Carnot-Carathéodory metric dc. It is proved that a compact set \(E\subset H\), satisfying an analog of P. Jones’ geometric lemma [“Rectifiable sets and the travelling salesman problem”, Invent. Math. 102, No. 1, 1–15 (1990; Zbl 0731.30018)] is contained in a rectifiable curve. The proof is given in terms of Heisenberg \(\beta\) numbers which measure set \(E\) is approximated by Heisenberg straight line.

MSC:
28A75 Length, area, volume, other geometric measure theory
43A80 Analysis on other specific Lie groups
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