Annales Academić Scientiarum Fennicć
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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