Constructing feasible open subsets for piecewise self-similar boundaries

Develop a general constructive method to find feasible open sets U^{(j)} ⊂ R^n for each self-similar boundary component F^{(j)} such that the projection condition S_i^{(j)} U^{(j)} ⊂ π_{F^{(j)}}^{-1}(S_i^{(j)} F^{(j)}), the domain-side projection condition U^{(j)} ⊂ π_{D}^{-1}(F^{(j)}), and disjointness U^{(j)} ∩ U^{(ℓ)} = ∅ for j ≠ ℓ hold simultaneously; in particular, provide general criteria or algorithms to construct “good feasible subsets” when V_c^{(j)} ∩ D is not itself feasible.

Background

To derive inner and outer fractal curvatures via tilings, the authors require feasible open sets U^{(j)} that satisfy projection and disjointness properties tailored to the domain D and its piecewise self-similar boundary ∂D = ⋃ F^{(j)}.

While central feasible sets V_c^{(j)} are known to satisfy projection conditions, ensuring feasibility and the additional domain-side constraints in general configurations is challenging. The authors explicitly state the lack of a general method to find such subsets.