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 $0<t<s\leq Q-1$. The visible boundary is the set of points that can be reached by a John curve from a fixed point $z_{0}\in \Omega$. This generalizes recent results by Koskela-Nandi-Nicolau (from $\mathbb{R}^2$) and Azzam ($\mathbb{R}^n$). In particular, our approach shows that the phenomenon is independent of the linear structure of the space.