  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.
