Density of polyhedral partitions

We prove the density of polyhedral partitions in the set of finite Caccioppoli partitions. Precisely, given a decomposition u of a bounded Lipschitz set Ω ⊂ R n into finitely many subsets of finite perimeter and ε > 0 , we prove that u is ε -close to a small deformation of a polyhedral decomposition v ε , in the sense that there is a C 1 diffeomorphism f ε : R n → R n which is ε -close to the identity and such that u ∘ f ε - v ε is ε -small in the strong BV norm. This implies that the energy of u is close to that of v ε for a large class of energies defined on partitions.