2006/05/16 by Raanan Schul, Schul, Raanan
Mathematics · #28A75 #Advanced Topology and Set Theory #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Metric Geometry (math.MG)
paper · pdf · doi:10.48550/arxiv.math/0605454
openalex publication_date 2006/05/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.