Accessible parts of the boundary for domains in metric measure spaces

Abstract

We prove in the setting of Q-Ahlfors regular PI-spaces the following result: if a domain has uniformly large boundary when measured with respect to the s-dimensional Hausdorff content, then its visible boundary has large t-dimensional Hausdorff content for every 0 < t < s ≤ Q − 1. The visible boundary is the set of points that can be reached by a John curve from a fixed point z 0 ∈ Ω. This generalizes recent results by Koskela-Nandi-Nicolau (from R 2 ) and Azzam (R n ). In particular, our approach shows that the phenomenon is independent of the linear structure of the space.