On the sum of a parallelotope and a zonotope
A parallelotope P is a polytope that admits a facet-to-facet tiling of space by translation copies of P along a lattice. The Voronoi cell P_V(L) of a lattice L is an example of a parallelotope. A parallelotope can be uniquely decomposed as the Minkowski sum of a zone closed parallelotope P and a zonotope Z(U), where Uis the set of vectors used to generate the zonotope. In this paper we consider the related question: When is the Minkowski sum of a general parallelotope and a zonotope P+Z(U) a parallelotope? Two necessary conditions are that the vectors of U have to be free and form a unimodular set. Given a unimodular set U of free vectors, we give several methods for checking if P+Z(U) is a parallelotope. Using this we classify such zonotopes for some highly symmetric lattices.
In the case of the root lattice E_6, it is possible to give a more geometric description of the admissible sets of vectors U. We found that the set of admissible vectors, called free vectors, is described by the well-known configuration of 27 lines in a cubic surface. Based on a detailed study of the geometry of PV(E6), we give a simple characterization of the configurations of vectors U such that P_V(E_6)+Z(U) is a parallelotope. The enumeration yields 10 maximal families of vectors, which are presented by their description as regular matroids.