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 ⊂ Rn 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 vε, in the sense that there is a C1 diffeomorphism fε: Rn→ Rn which is ε-close to the identity and such that u∘ fε- vε is ε-small in the strong BV norm. This implies that the energy of u is close to that of vε for a large class of energies defined on partitions.