TY - JOUR

T1 - Voronoi’s Conjecture for extensions of Voronoi parallelohedra

AU - Magazinov, Alexander

N1 - Publisher Copyright:
© 2015, Mathematical Sciences Publishers. All rights reserved.

PY - 2015

Y1 - 2015

N2 - 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.

AB - 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.

KW - Voronoi’s Conjecture

KW - free segment

KW - parallelohedron

KW - reducible parallelohedron

KW - tiling

UR - http://www.scopus.com/inward/record.url?scp=85052712887&partnerID=8YFLogxK

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AN - SCOPUS:85052712887

SN - 2220-5438

VL - 5

SP - 86

EP - 131

JO - Moscow Journal of Combinatorics and Number Theory

JF - Moscow Journal of Combinatorics and Number Theory

IS - 3

ER -