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 $D_{2m}^+$, $m\geqslant 4$, we prove that their Voronoi polyhedra are nonfree in all directions.