Ahlfors-regular curves in metric spaces

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.