*Preprint*

**Inserted:** 28 jun 2019

**Last Updated:** 1 jul 2019

**Year:** 2019

**Abstract:**

We show a sufficient criterion to determine if a planar set $\Omega$ is a minimizer of the prescribed curvature functional among all of its subsets. 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. Specifically, if a Jordan domain $\Omega$ possesses the interior disk property of radius $

\Omega

/P(\Omega)$, then it is a self-Cheeger set; if it possesses the strict interior disk property then it is a minimal Cheeger set, i.e. the unique minimizer. As a side effect we provide a way to build self-Cheeger sets.

**Keywords:**
prescribed mean curvature, inner Cheeger formula, Cheeger costant, self-Cheeger sets, perimeter minimizer

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