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 $0<t<s\le Q-1$. The visible boundary is the set of points that can be reached by a John curve from a fixed point $z_0\in \mathrm{\Omega }$. 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.