Revista Matemática Iberoamericana

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Volume 23, Issue 2, 2007, pp. 437–480
DOI: 10.4171/RMI/502

Published online: 2007-08-31

The Geometric Traveling Salesman Problem in the Heisenberg Group

Fausto Ferrari[1], Bruno Franchi[2] and Hervé Pajot[3]

(1) Università di Bologna, Italy
(2) Università di Bologna, Italy
(3) Université Grenoble I, Saint-Martin-d'Hères, France

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

Keywords: Heisenberg group, Carnot-Carath´eodory metric, rectifiable curve, Traveling Salesman Problem

Ferrari Fausto, Franchi Bruno, Pajot Hervé: The Geometric Traveling Salesman Problem in the Heisenberg Group. Rev. Mat. Iberoam. 23 (2007), 437-480. doi: 10.4171/RMI/502