Lifting of parallelohedra

Abstract

Abstract A parallelohedron is a polyhedron that can tessellate the space via translations without gaps and overlaps. Voronoi conjectured that any parallelohedron is affinely equivalent to a Dirichlet-Voronoi cell of some lattice. Delaunay used the term displacement parallelohedron in his paper “Sur la tiling régulière de l’espace à 4 dimensions. Première partie”, where the four-dimensional parallelohedra are listed. In our work, such a parallelohedron is called a lifted parallelohedron, since it is obtained as an extension of a parallelohedron to a parallelohedron of dimension larger by one. It is shown that the operation of lifting yields precisely parallelohedra whose Minkowski sum with some nontrivial segment is again a parallelohedron. It is proved that Voronoi’s conjecture holds for parallelohedra admitting lifts and lifted in general position. Bibliography: 20 titles.