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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

Citations:

Zbl 0731.30018
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References:

[1] Ambrosio, L., Rigot, S.: Optimal mass transportation in the Heisenberg group. J. Funct. Anal. 208 (2004), no. 2, 261-301. · Zbl 1076.49023 · doi:10.1016/S0022-1236(03)00019-3
[2] Ambrosio, L., Tilli, P.: Topics on analysis in metric spaces . Cambridge University Press, 2004. · Zbl 1080.28001
[3] Bellaï che, A.: The tangent space in subriemannian geometry. In Subriemannian Geometry , 1-78. Progr. Math. 144 . Birkhäuser, Basel, 1996. · Zbl 0862.53031
[4] Bishop, C., Jones, P.: Harmonic measure, \(L^2\) estimates and the Schwarzian derivative. J. Anal. Math. 62 (1994), 77-113. · Zbl 0801.30024 · doi:10.1007/BF02835949
[5] Bishop, C., Jones, P.: Hausdorff dimension and Kleinian groups. Acta Math. 179 (1997), no. 1, 1-39. · Zbl 0921.30032 · doi:10.1007/BF02392718
[6] Bishop, C., Jones, P., Pemantle, R., Peres, Y.: The dimension of the Brownian frontier is greater than \(1\). J. Funct. Anal. 43 (1997), no. 2, 309-336. · Zbl 0870.60077 · doi:10.1006/jfan.1996.2928
[7] Blumenthal, L. M., Menger, K.: Studies in geometry . W. H. Freeman and Company, San Francisco, 1970. · Zbl 0204.53401
[8] Burago, D., Burago, Y., Ivanov, S.: A course in metric geometry . Graduate Studies in Mathematics 33 . American Mathematical Society, Providence, 2001. · Zbl 0981.51016
[9] David, G., Semmes, S.: Analysis of and on uniformly rectifiable sets . Mathematical Surveys and Monographs 38 . American Mathematical Society, Providence, 1993. · Zbl 0832.42008
[10] Federer, H.: Geometric measure theory . Die Grundlehren der mathematischen Wissenschaften 153 . Springer-Verlag, New York, 1969. · Zbl 0176.00801
[11] Franchi, B., Serapioni, R., Serra Cassano, F.: Rectifiability and perimeter in the Heisenberg group. Math. Ann. 321 (2001), no. 3, 479-531. · Zbl 1057.49032 · doi:10.1007/s002080100228
[12] Franchi, B., Serapioni, R., Serra Cassano, F.: Regular submanifolds, graphs and area formula in Heisenberg groups. Adv. Math. 211 (2007), 152-203. · Zbl 1125.28002 · doi:10.1016/j.aim.2006.07.015
[13] Gaveau, B.: Principe de moindre action, propagation de la chaleur et estimées sous elliptiques sur certains groupes nilpotents. Acta Math. 139 (1977), no. 1-2, 95-153. · Zbl 0366.22010 · doi:10.1007/BF02392235
[14] Graczyk, J., Jones, P.: Dimension of the boundary of quasi-conformal Siegel disks. Invent. Math. 148 (2002), no. 3, 465-493. · Zbl 1079.37507 · doi:10.1007/s002220100198
[15] Gromov, M.: Metric structures for Riemannian and non-Riemannian spaces . Progress in Mathematics 152 . Birkhäuser, Boston, MA, 1999. · Zbl 0953.53002
[16] Hahlomaa, I.: Menger curvature and Lipschitz parametrizations in metric spaces. Fund. Math. 185 (2005), no. 2, 143-169. · Zbl 1077.54016 · doi:10.4064/fm185-2-3
[17] Jones, P.: Square functions, Cauchy integrals, analytic capacity, and harmonic measure. In Harmonic analysis and partial differential equations , 24-68. Lecture Notes in Mathematics 1384 . Springer-Verlag, 1989. · Zbl 0675.30029 · doi:10.1007/BFb0086793
[18] Jones, P.: Rectifiable sets and the traveling salesman problem. Invent. Math. 102 (1990), no. 1, 1-15. · Zbl 0731.30018 · doi:10.1007/BF01233418
[19] Mattila, P.: Geometry of sets and measures in Euclidean spaces . Cambridge Studies in Advanced Mathematics 44 . Cambridge University Press, Cambridge, 1995. · Zbl 0819.28004
[20] Mitchell, J.: On Carnot-Carathéodory metrics. J. Differential Geom. 21 (1985), no. 1, 35-45. · Zbl 0554.53023
[21] Okikiolu, K.: Characterizations of subsets of rectifiable curves in \(\R^n\). J. London Math. Soc. (2) 46 (1992), no. 2, 336-348. · Zbl 0758.57020 · doi:10.1112/jlms/s2-46.2.336
[22] Pajot, H.: Sous-ensembles de courbes Ahlfors-régulières et nombre de Jones. Publ. Mat. 40 (1996), no. 2, 497-526. · Zbl 0933.28002 · doi:10.5565/PUBLMAT_40296_17
[23] Pajot, H.: Analytic capacity, rectifiability, Menger curvature and the Cauchy integral . Lecture Notes in Mathematics 1799 . Springer Verlag, Berlin, 2002. · Zbl 1043.28002
[24] Pansu, P.: Géométrie du Groupe d’Heisenberg . Thèse pour le titre de Docteur de 3ème cycle, Université Paris VII, 1982.
[25] Pansu, P.: Une inégalité isopérimétrique sur le groupe de Heisenberg. C. R. Acad. Sci. Paris Sér. I Math. 295 (1982), no. 2, 127-130. · Zbl 0502.53039
[26] Schul, R.: Subset of rectifiable curves in Hilbert space and the analyst’s TSP . PhD dissertation, Yale University, 2005. · Zbl 1152.28006
[27] Stein, E. M.: Harmonic Analysis: real-variable methods, orthogonality, and oscillatory integrals. Princeton Mathematical Series 43 . Princeton University Press, Princeton, 1993. · Zbl 0821.42001
[28] Varopoulos, N. Th., Saloff-Coste, L., Coulhon, T.: Analysis and Geometry on Groups . Cambridge Tracts in Mathematics 100 . Cambridge University Press, Cambridge, 1992. · Zbl 0813.22003
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