Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 42, 2017, 649-692

THE ANALYST'S TRAVELING SALESMAN THEOREM IN GRAPH INVERSE LIMITS

Guy C. David and Raanan Schul

New York University, Courant Institute of Mathematical Sciences
New York, NY 10012, U.S.A.; guydavid 'at' math.nyu.edu

Stony Brook University, Department of Mathematics
Stony Brook, NY 11794-3651, U.S.A.; schul 'at' math.sunysb.edu

Abstract. We prove a version of Peter Jones' analyst's traveling salesman theorem in a class of highly non-Euclidean metric spaces introduced by Laakso and generalized by Cheeger–Kleiner. These spaces are constructed as inverse limits of metric graphs, and include examples which are doubling and have a Poincaré inequality. We show that a set in one of these spaces is contained in a rectifiable curve if and only if it is quantitatively “flat” at most locations and scales, where flatness is measured with respect to so-called monotone geodesics. This provides a first examination of quantitative rectifiability within these spaces.

2010 Mathematics Subject Classification: Primary 28A75.

Key words: Beta numbers, metric space, traveling salesman, curvature.

Reference to this article: : The analyst's traveling salesman theorem in graph inverse limits. Ann. Acad. Sci. Fenn. Math. 42 (2017), 649-692.

Full document as PDF file

https://doi.org/10.5186/aasfm.2017.4260