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

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