The Voronoi conjecture for parallelohedra with simply connected $\delta$-surface

Abstract

We show that the Voronoi conjecture is true for parallelohedra with simply connected $\delta$-surface. Namely, we show that if the boundary of parallelohedron $P$ remains simply connected after removing closed non-primitive faces of codimension 2, then $P$ is affinely equivalent to a Dirichlet-Voronoi domain of some lattice. Also we construct the $\pi$-surface associated with a parallelohedron and give another condition in terms of homology group of the constructed surface. Every parallelohedron with simply connected $\delta$-surface also satisfies the condition on homology group of the $\pi$-surface.

13 pages, 4 figures

References in corpus (1)

• On π-surfaces of four-dimensional parallelohedra

Cited by in corpus (6)

• Salem numbers in dynamics of Kähler threefolds and complex tori
• Voronoi conjecture for five-dimensional parallelohedra
• On π-surfaces of four-dimensional parallelohedra
• On the Voronoi Conjecture for combinatorially Voronoi parallelohedra in dimension five
• On combinatorics of Voronoi polytopes for perturbations of the dual root lattices
• Tight fans and their canonical scalings