Ahlfors-Regular Curves In Metric Spaces

Details Ahlfors-Regular Curves In Metric Spaces Ahlfors-Regular Curves In Metric Spaces

2006-05-16

arxiv.org/abs/math/0605454v2

Mathematics

Metric Geometry

25 pages, No figures. revisions after referee comments. to appear in Annales Academae Scientiarum Fennicae Mathematica

Scientific paper

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 I. 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 P. Jones, K. Okikiolu, and G. David and S. Semmes, and it can be stated in terms of average Menger curvature.

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