Free and nonfree Voronoi polyhedra

The Voronoi polyhedron of some point v of a translation lattice is the closure of the set of points in space that are closer to v than to any other lattice points. Voronoi polyhedra are a special case of parallelohedra, i.e., polyhedra whose parallel translates can fill the entire space without gaps and common interior points. The Minkowski sum of a parallelohedron with a segment is not always a parallelohedron. A parallelohedron P is said to be free along a vector e if the sum of P with a segment of the line spanned by e is a parallelohedron. We prove a theorem stating that if the Voronoi polyhedron P _v (f) of a quadratic form f is free along some vector, then the Voronoi polyhedron P _v (g) of each form g lying in the closure of the L-domain of f is also free along some vector. For the dual root lattice E ₆*, and the infinite series of lattices D _2m ⁺ , m ≥ 4, we prove that their Voronoi polyhedra are nonfree in all directions.