Annales Academię Scientiarum Fennicę

Mathematica

Volumen 32, 2007, 437-460

# AHLFORS-REGULAR CURVES IN METRIC SPACES

## Raanan Schul

UCLA, University of California, Department of Mathematics

Box 95155, Los Angeles, CA 90095-1555, U.S.A.; schul 'at' math.ucla.edu

**Abstract.**
We discuss 1-Ahlfors-regular connected sets
in a general metric space and prove that such sets are
'flat' on most scales
and in most locations. Our result is quantitative,
and when combined with work of Hahlomaa, gives a
characterization of 1-Ahlfors regular subsets of 1-Ahlfors-regular
curves in metric spaces. Our result is a generalization to the metric space
setting of the analyst's (geometric) traveling salesman theorems of Jones,
Okikiolu, and David and Semmes,
and it can be stated in terms of average Menger curvature.

**2000 Mathematics Subject Classification:**
Primary 28A75; Secondary 51F99.

**Key words:**
Menger curvature, Hausdorff length, Ahlfors
regular, travelling salesman, Jones beta number.

**Reference to this article:** R. Schul:
Ahlfors-regular curves in metric spaces.
Ann. Acad. Sci. Fenn. Math. 32 (2007), 437-460.

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