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Journal of Convex Analysis 28 (2021), No. 3, 951--958
Copyright Heldermann Verlag 2021

A Sufficient Criterion to Determine Planar Self-Cheeger Sets

Giorgio Saracco
Scuola Internazionale Superiore di Studi Avanzati, Trieste, Italy


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.

Keywords: Cheeger constant, inner Cheeger formula, self-Cheeger sets, perimeter minimizer, prescribed mean curvature.

MSC: 49Q10; 35J93, 49Q20.

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