  Given an open, bounded, planar set $\Omega$, we consider its $p$-Cheeger sets
and its isoperimetric sets. We study the set-valued map
$\mathfrak{V}:[\frac12,+\infty)\rightarrow\mathcal{P}((0,|\Omega|])$
associating to each $p$ the set of volumes of $p$-Cheeger sets. We show that
whenever $\Omega$ satisfies some geometric structural assumptions (convex sets
are encompassed), the map is injective, and continuous in terms of
$\Gamma$-convergence. Moreover, when restricted to $(\frac 12, 1)$ 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.
