An upper bound for the length of a traveling salesman path in the Heisenberg group. (English) Zbl 1355.28005
The authors show that a sufficient condition for a subset E in the Heisenberg group (endowed with the Carnot-Carathéodory metric) to be contained in a rectifiable curve is that it satisfies a modified analogue of Peter Jones’ geometric lemma. The estimates given in this paper improve those of [F. Ferrari et al., Rev. Mat. Iberoam. 23, No. 2, 437–480 (2007; Zbl 1142.28004)], allowing for any power \(r<4\) to replace the power 2 of the Jones-\(\beta\)-number. This complements in an open ended way the work of the authors in [Trans. Am. Math. Soc. 368, No. 7, 4585–4620 (2016; Zbl 1350.53044)], where they showed that such an estimate is necessary, however with the power \(r=4\).
Reviewer: Peibiao Zhao (Nanjing)
Keywords:
metric geometryReferences:
| [1] | Azzam, J. and Schul, R.: How to take shortcuts in Euclidean space: making a given set into a short quasi-convex set. Proc. London Math. Soc.105 (2012), no. 2, 367–392. · Zbl 1250.28001 |
| [2] | Balogh, Z. M., Hoefer-Isenegger, R. and Tyson, J.: Lifts of Lipschitz maps and horizontal fractals in the Heisenberg group. Ergodic Theory Dyn. Syst.26 (2006), no. 3, 621–651. · Zbl 1097.22005 |
| [3] | Cygan, H.: Subadditivity of homogeneous norms on certain nilpotent Lie groups. Proc. Amer. Math. Soc.83 (1981), no. 1, 69–70. · Zbl 0475.43010 |
| [4] | David, G. and Semmes, S.: Analysis of and on uniformly rectifiable sets. Mathematical Surveys and Monographs 38, American Mathematical Society, 1993. · Zbl 0832.42008 |
| [5] | David, G. and Semmes, S.: Singular integrals and rectifiable sets inRn: beyond Lipschitz graphs. Ast’erique 193, 1991. |
| [6] | Franchi, B., Ferrari, F. and Pajot, H.: The geometric traveling salesman problem in the Heisenberg group. Rev. Mat. Iberoam.23 (2007), no. 2, 437–480. · Zbl 1142.28004 |
| [7] | Hahlomaa, I.: Menger curvature and Lipschitz parametrizations in metric spaces. Fund. Math.185 (2005), no. 2, 143–169. · Zbl 1077.54016 |
| [8] | Hahlomaa, I.: Curvature integral and Lipschitz parametrizations in 1-regular metric spaces. Ann. Acad. Sci. Fenn. Math.32 (2007), 99–123. Upper bound for the length of TSP in the Heisenberg group417 · Zbl 1117.28001 |
| [9] | Jones, P. W.: Rectifiable sets and the traveling salesman problem. Invent. Math. 102 (1990), no. 1, 1–15. · Zbl 0731.30018 |
| [10] | Juillet, N.: A counterexample for the geometric traveling salesman problem in the Heisenberg group. Rev. Mat. Iberoam.26 (2010), no. 3, 1035–1056. · Zbl 1206.28003 |
| [11] | Li, S.: Coarse differentiation and quantitative nonembeddability for Carnot groups. J. Funct. Anal.266 (2014), 4616–4704. · Zbl 1311.46021 |
| [12] | Li, S.: Markov convexity and nonembeddility of the Heisenberg group. Ann. Inst. Fourier66 (2016), no. 4, 1615–1651. · Zbl 1439.30087 |
| [13] | Li, S. and Schul, R.: The traveling salesman problem in the Heisenberg group: upper curvature bound. Trans. Amer. Math. Soc.368 (2016), no. 7, 4585–4620. · Zbl 1350.53044 |
| [14] | Montgomery, R.: A tour of sub-riemannian geometries, their geodesics and applications. Mathematical Surveys and Monographs 91, American Mathematical Society, 2002. · Zbl 1044.53022 |
| [15] | Okikiolu, K.: Characterizations of subsets of rectifiable curves inRn. J. London Math. Soc. (2)46 (1992), 336–348. · Zbl 0758.57020 |
| [16] | Pajot, H.: Analytic capacity, rectifiability, Menger curvature and the Cauchy integral. Lecture Notes in Mathematics 1799, Springer-Verlag, 2002. · Zbl 1043.28002 |
| [17] | Schul, R.: Ahlfors-regular curves in metric spaces. Ann. Acad. Sci. Fenn. Mat.32 (2007), 437–460. · Zbl 1122.28006 |
| [18] | Schul, R.: Subsets of rectifiable curves in Hilbert space – the analyst’s TSP. J. Anal. Math.103 (2007), 331–375. · Zbl 1152.28006 |
| [19] | Schul, R.: Analyst’s traveling salesman theorems. A survey. In The tradition of Ahlfors-Bers IV, 209–220. Contemp. Math. 432, Amer. Math. Soc., Providence, RI, 2007. · Zbl 1187.49039 |
| [20] | Tolsa, X.: Analytic capacity, the Cauchy transform, and non-homogeneous Calder’on–Zygmund theory. Progress in Mathematics 307, Birkh”auser/Springer, Cham, 2014. Received April 11, 2014; revised March 29, 2015. Sean Li: Department of Mathematics, The University of Chicago, 5734 S University Avenue, Chicago, IL 60637, USA. E-mail:seanli@math.uchicago.edu Raanan Schul: Department of Mathematics, Stony Brook University, Stony Brook, NY 11794-3651, USA. E-mail:schul@math.sunysb.edu S. Li was partially supported by a postdoctoral research fellowship NSF DMS-1303910. R. Schul was partially supported by a fellowship from the Alfred P. Sloan Foundation as well as by NSF DMS 11-00008. IntroductionPreliminariesLemmas: future ballsMain propositionThe construction and its length |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
