  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.
