Self-affine tiling of polyhedra

We obtain a complete classification of polyhedral sets (unions of finitely many convex polyhedra) that admit self-affine tilings, i.e., partitions into parallel shifts of one set that is affinely similar to the initial one. In every dimension, there exist infinitely many nonequivalent polyhedral sets possessing this property. Under an additional assumption that the affine similarity is defined by an integer matrix and by integer shifts (“digits”) from different quotient classes with respect to this matrix, the only polyhedral set of this kind is a parallelepiped. Applications to multivariate wavelets and to Haar systems are discussed.