Open AccessAccessible parts of the boundary for domains in metric measure spaces
Author(s) -
Ryan Gibara,
Riikka Korte
Publication year - 2022
Publication title -
annales fennici mathematici
Language(s) - English
Resource type - Journals
eISSN - 2737-114X
pISSN - 2737-0690
DOI - 10.54330/afm.116365
Subject(s) - mathematics , boundary (topology) , hausdorff measure , hausdorff space , domain (mathematical analysis) , measure (data warehouse) , omega , metric (unit) , metric space , space (punctuation) , mathematical analysis , combinatorics , pure mathematics , hausdorff dimension , discrete mathematics , physics , computer science , operations management , database , quantum mechanics , economics , operating system
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.