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

Weighted Cheeger sets are domains of isoperimetry