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Mat. Sb., 2019, Volume 210, Number 10, Pages 99–121 (Mi msb8871)  

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.
Author to whom correspondence should be addressed

DOI: https://doi.org/10.4213/sm8871

Full text: PDF file (700 kB)
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English version:
Sbornik: Mathematics, 2019, 210:10, 1434–1455

Bibliographic databases:

UDC: 511.5+514.174.6
MSC: Primary 52B11; Secondary 52C22
Received: 29.11.2016 and 09.04.2019

Citation: V. P. Grishukhin, V. I. Danilov, “Lifting of parallelohedra”, Mat. Sb., 210:10 (2019), 99–121; Sb. Math., 210:10 (2019), 1434–1455

Citation in format AMSBIB
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