*Submitted Paper*

**Inserted:** 9 oct 2016

**Last Updated:** 4 aug 2017

**Year:** 2016

**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 $\mathcal{H}^{n-1}(A^{(1)} \cap \partial A)=0$ satisfies a relative isoperimetric inequality. If $\Omega$ itself is a connected minimizer such that $\mathcal{H}^{n-1}(\Omega^{(1)} \cap \partial \Omega)=0$, 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 $\mathcal{H}^{n-1}(\Omega^{(1)} \cap \partial \Omega)=0$.

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