Accessible parts of the boundary for domains in metric measure spaces

TL;DR

This paper demonstrates that in metric measure spaces with certain regularity, the size of the boundary influences the size of the visible boundary, extending previous Euclidean results to more general spaces.

Contribution

It generalizes boundary visibility results from Euclidean spaces to Q-Ahlfors regular PI-spaces, showing independence from linear structure.

Findings

Large boundary implies large visible boundary in Hausdorff content

Results extend to general metric measure spaces, not just Euclidean

Applicable to domains with uniformly large boundary measure

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.