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.
F. Ferrari, B. Franchi, H. Pajot (2007). The Geometric Traveling Salesman Problem in the Heisenberg Group. REVISTA MATEMATICA IBEROAMERICANA, 23, 437-480.
The Geometric Traveling Salesman Problem in the Heisenberg Group
FERRARI, FAUSTO;FRANCHI, BRUNO;
2007
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.File in questo prodotto:
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