The Geometric Traveling Salesman Problem in the Heisenberg Group

Abstract

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.