Self-similarity, positive Lebesgue measure and nonempty interior

In this paper, we introduce BBI spaces (“big balls of itself”), which based on the notion of BPI spaces (“big pieces of itself”) used by David and Semmes to study self-similarity. We prove that the “self-similar” construction described by BBI spaces ensures the equivalence of positive Lebesgue measure and nonempty interior. We apply this result to self-conformal sets satisfying the WSC and prove that positive Lebesgue measure implies nonempty interior for such sets. This generalizes Zerner's corresponding result for self-similar sets.