Ahlfors-Regular Curves In Metric Spaces

• DOI: 10.48550/arxiv.math/0605454
• Publication Date: 2006-01-01
• Authors: Raanan Schul
• Keywords: Characterization

We discuss 1-Ahlfors-regular connected sets in a general metric space and prove that such are `flat' on most scales locations. Our result is quantitative, when combined with work of I. Hahlomaa, gives characterization 1-Ahlfors regular subsets curves spaces. generalization to the setting Analyst's (Geometric) Traveling Salesman theorems P. Jones, K. Okikiolu, G. David S. Semmes, it can be stated terms average Menger curvature.