Parallelohedra defined by quadratic forms

The results of Section III of G. F. Voronoi's famous memoir are presented in modern terms. The description of a parallelohedron by a system of linear constraints with quadratic right-hand side naturally leads to the notion of a contact face, which is called a standard face by N. P. Dolbilin. It is proved that a nonempty intersection of two contact faces generates a $4$- or $6$-belt of these contact faces. As an example, zonotopes defined by quadratic forms are considered. In particular, zonotopal parallelohedra of Voronoi's principal domain are examined in detail. It is shown that these parallelohedra are submodular polytopes, which are frequently encountered in combinatorial theory.