*Published Paper*

**Inserted:** 28 jun 2019

**Last Updated:** 21 sep 2021

**Journal:** J. Convex Anal.

**Volume:** 28

**Number:** 3

**Pages:** 951--958

**Year:** 2021

**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 $P(E)/\textrm{vol}(E)$ among all of its subsets. Specifically, if a Jordan domain $\Omega$ possesses the interior disk property of radius $\textrm{vol}(\Omega)/P(\Omega)$, 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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