Density of polyhedral partitions

We prove the density of polyhedral partitions in the set of finite Caccioppoli partitions. Precisely, we consider a decomposition $u$ of a bounded Lipschitz set $\Omega\subset\mathbb R^n$ into finitely many subsets of finite perimeter, which can be identified with a function in $SBV_{\rm loc}(\Omega;{\cal Z})$ with ${\cal Z}\subset \mathbb R^N$ a finite set of parameters. For all $\varepsilon>0$ we prove that such a $u$ is $\varepsilon$-close to a small deformation of a polyhedral decomposition $v_\varepsilon$, in the sense that there is a $C^1$ diffeomorphism $f_\varepsilon:\mathbb R^n\to\mathbb R^n$ which is $\varepsilon$-close to the identity and such that $u\circ f_\varepsilon-v_\varepsilon$ is $\varepsilon$-small in the strong $BV$ norm. This implies that the energy of $u$ is close to that of $v_\varepsilon$ for a large class of energies defined on partitions. Such type of approximations are very useful in order to simplify computations in the estimates of $\Gamma$-limits.