A Sufficient Criterion to Determine Planar Self-Cheeger Sets

We prove a sufficient criterion to determine if a planar set $\Omega$ minimizes the prescribed curvature functional $\mathcal{F}_\kappa[E]:=P(E)-\kappa|E|$ amongst $E\subset \Omega$. As a special case, we derive a sufficient criterion to determine if $\Omega$ is a self-Cheeger set, i.e. if it minimizes the ratio $P(E)/|E|$ among all of its subsets. As a side effect we provide a way to build self-Cheeger sets.