*Online first*

**Inserted:** 9 oct 2016

**Last Updated:** 12 sep 2017

**Journal:** Manuscripta Math.

**Pages:** 1-11

**Year:** 2017

**Doi:** 10.1007/s00229-017-0974-z

**Abstract:**

We consider a generalization of the Cheeger problem in a bounded, open set
$\Omega$ by replacing the perimeter functional with a Finsler-type surface
energy and the volume with suitable powers of a weighted volume. We show that
any connected minimizer $A$ of this weighted Cheeger problem such that
$H^{n-1}(A^{(1)} \cap \partial A)=0$ satisfies a relative isoperimetric
inequality. If $\Omega$ itself is a connected minimizer such that
$H^{n-1}(\Omega^{(1)} \cap \partial \Omega)=0$, then it allows the classical
Sobolev and $BV$ embeddings and the classical $BV$ trace theorem. The same
result holds for any connected minimizer whenever the weights grant the
regularity of perimeter-minimizer sets and $\Omega$ is such that $

\partial
\Omega

=0$ and $H^{n-1}(\Omega^{(1)} \cap \partial \Omega)=0$.

**Keywords:**
Cheeger problem, Sobolev embeddings, trace theorems

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