V. P. Grishukhin, V. I. Danilov, “Lifting of parallelohedra”, Sb. Math., 210:10 (2019), 1434–1455 (original) (raw)

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Lifting of parallelohedra

V. P. Grishukhin, V. I. Danilov

Central Economics and Mathematics Institute of Russian Academy of Sciences, Moscow, Russia

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.

Keywords: parallelohedral tiling, lattice, free direction, generatrissa, lamina.

Received: 29.11.2016 and 09.04.2019

Bibliographic databases:

Document Type: Article

UDC: 511.5+514.174.6

Language: English

Original paper language: Russian

Citation: V. P. Grishukhin, V. I. Danilov, “Lifting of parallelohedra”, Sb. Math., 210:10 (2019),

1434–1455

Citation in format AMSBIB

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