Ahlfors-regular Curves in Metric Spaces
نویسنده
چکیده
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 1Ahlfors-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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ar X iv : m at h / 06 05 45 4 v 1 [ m at h . M G ] 1 6 M ay 2 00 6 Ahlfors - Regular Curves In Metric Spaces
We discuss 1-Ahlfors-regular connected sets in a metric space. We prove that such a set is ‘flat’ on most scales and locations. We give a quantitative version of this. This, together with work of I. Hahlomaa, gives a characterization of 1-Ahlfors regular subsets of 1-Ahlfors-regular curves in a metric space, generalizing in a way the Analyst’s (Geometric) Traveling Salesman theorems by P. Jones...
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