Isoperimetric sets and p-Cheeger sets are in bijection

Given an open, bounded, planar set Ω , we consider its p-Cheeger sets and its isoperimetric sets. We study the set-valued map V: [1 / 2 , + ∞) → P((0 , | Ω |]) associating to each p the set of volumes of p-Cheeger sets. We show that whenever Ω satisfies some geometric structural assumptions (convex sets are encompassed), the map is injective, and continuous in terms of Γ -convergence. Moreover, when restricted to (1 / 2 , 1) such a map is univalued and is in bijection with its image. As a consequence of our analysis we derive some fine boundary regularity result.