Naslov
On the Voronoi Conjecture for combinatorially Voronoi parallelohedra

Autori
Dutour Sikirić, Mathieu ; Garber, Alexey ; Magazinov, Alexander

Izvornik
SIAM journal on discrete mathematics (0895-4801) 34 (2020), 4; 2481-2501

Vrsta, podvrsta i kategorija rada
Radovi u časopisima, članak, znanstveni

Ključne riječi
Voronoi Conjecture ; Voronoi polytope ; enumeration

Sažetak
In a recent paper Garber, Gavrilyuk and Magazinov proposed a sufficient combinatorial condition for a parallelohedron to be affinely Voronoi. We show that this condition holds for all five-dimensional Voronoi parallelohedra. Consequently, the Voronoi conjecture in R^5 holds if and only if every five-dimensional parallelohedron is combinatorially Voronoi. Here, by saying that a parallelohedron P is combinatorial Voronoi, we mean that the tiling T(P) by translates of P is combinatorially isomorphic to some tiling T(P'), where P' is a Voronoi parallelohedron, and that the isomorphism naturally induces a linear isomorphism of lattices Lambda(P) and Lambda(P'). We also propose a new sufficient condition implying that a parallelohedron is affinely Voronoi. The condition is based on the new notion of the Venkov complex associated with a parallelohedron.

Izvorni jezik
Engleski

Znanstvena područja
Matematika

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