Given a planar, open set $\Omega$, satisfying some weak regularity assumptions, we will consider the following three different problems among subsets of $\Omega$: the isoperimetric problem for volume $V$; the prescribed curvature problem for curvature $\kappa$; the $p$-Cheeger problem.

We shall see that there exist bijections $\mathfrak{K}$ and $\mathfrak{V}$ between these problems in the following sense: a set is isoperimetric of volume $V$ if and only if it attains the minimum of the prescribed curvature functional for curvature $\mathfrak{K}(V)$; analogously a set is $p$-Cheeger if and only if it is isoperimetric of volume $\mathfrak{V}(p)$.

As a byproduct we infer some convexity properties on the isoperimetric profile, and some fine regularity properties on the contact surface of minimizers.

Based on joint works with Caroccia, Leonardi, Neumayer, and Pratelli.