Almost uniform domains and Poincaré inequalities

Here we show existence of many subsets of Euclidean spaces that, despite having empty interior, still support Poincaré inequalities with respect to the restricted Lebesgue measure. Most importantly, despite the explicit constructions in our proofs, our methods do not depend on any rectilinear or self‐similar structure of the underlying space. We instead employ the uniform domain condition of Martio and Sarvas. This condition relies on the measure density of such subsets, as well as the regularity and relative separation of their boundary components.