Voronoi’s Conjecture for extensions of Voronoi parallelohedra

Alexander Magazinov*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


In 1908 Voronoi conjectured that every parallelohedron is a a Voronoi parallelohedron for some Euclidean metric in Ed. Although the conjecture is still neither proved, nor disproved, there are several positive results for some special classes of parallelohedra. In this paper we extend the list of such classes by one new case. Let I be a segment in the d-dimensional Euclidean space Ed. Let P and P + I be parallelohedra in Ed, where the plus sign denotes the Minkowski sum. We prove that, if Voronoi’s Conjecture holds for P, then Voronoi’s Conjecture holds for P + I as well.

Original languageEnglish
Pages (from-to)86-131
Number of pages46
JournalMoscow Journal of Combinatorics and Number Theory
Issue number3
StatePublished - 2015
Externally publishedYes


  • Voronoi’s Conjecture
  • free segment
  • parallelohedron
  • reducible parallelohedron
  • tiling


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