Ahlfors-Regular Curves In Metric Spaces
Metric Geometry
2007-05-23 v2 Classical Analysis and ODEs
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 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.
Cite
@article{arxiv.math/0605454,
title = {Ahlfors-Regular Curves In Metric Spaces},
author = {Raanan Schul},
journal= {arXiv preprint arXiv:math/0605454},
year = {2007}
}
Comments
25 pages, No figures. revisions after referee comments. to appear in Annales Academae Scientiarum Fennicae Mathematica