Isoperimetric sets and $p$-Cheeger sets are in bijection

Given an open, bounded, planar set $\mathrm{\Omega }$, we consider its $p$-Cheeger sets and its isoperimetric sets. We study the set-valued map $\mathfrak{V}:[\frac{1}{2},+\mathrm{\infty })\to \mathcal{P}((0,\mathrm{\mid }\mathrm{\Omega }\mathrm{\mid }])$ associating to each $p$ the set of volumes of $p$-Cheeger sets. We show that whenever $\mathrm{\Omega }$ satisfies some geometric structural assumptions (convex sets are encompassed), the map is injective, and continuous in terms of $\mathrm{\Gamma }$-convergence. Moreover, when restricted to $(\frac{1}{2},1)$ 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.