47 research outputs found
Voronoi conjecture for five-dimensional parallelohedra
We prove the Voronoi conjecture for five-dimensional parallelohedra. Namely,
we show that if a convex five-dimensional polytope tiles
with translations, then is an affine image of the Dirichlet-Voronoi cell
for a five-dimensional lattice
The complete classification of five-dimensional Dirichlet-Voronoi polyhedra of translational lattices
In this paper we report on the full classification of Dirichlet-Voronoi
polyhedra and Delaunay subdivisions of five-dimensional translational lattices.
We obtain a complete list of affine types (L-types) of Delaunay
subdivisions and it turns out that they are all combinatorially inequivalent,
giving the same number of combinatorial types of Dirichlet-Voronoi polyhedra.
Using a refinement of corresponding secondary cones, we obtain
contraction types. We report on details of our computer assisted enumeration,
which we verified by three independent implementations and a topological mass
formula check.Comment: 16 page
VORONOI CONJECTURE FOR FIVE-DIMENSIONAL PARALLELOHEDRA
We prove the Voronoi conjecture for five-dimensional parallelohedra. Namely, we show that if a convex five-dimensional polytope P tiles R5 with translations, then P is an affine image of the Dirichlet-Voronoi cell for a five-dimensional lattice
Zonotopes, Dicings, and Voronoi’s Conjecture on Parallelohedra
AbstractIn 1909, Voronoi conjectured that if some selection of translates of a polytope forms a facet-to-facet tiling of euclidean space, then the polytope is affinely equivalent to the Voronoi polytope for a lattice. He referred to polytopes with this tiling property as parallelohedra, but they are now frequently called parallelotopes. I show that Voronoi’s conjecture holds for the special case where the parallelotope is a zonotope. I also show that the Voronoi polytope for a lattice is a zonotope if and only if the Delaunay tiling for the lattice is a dicing (defined at the beginning of Section 3)
On the density of sets avoiding parallelohedron distance 1
The maximal density of a measurable subset of R^n avoiding Euclidean
distance1 is unknown except in the trivial case of dimension 1. In this paper,
we consider thecase of a distance associated to a polytope that tiles space,
where it is likely that the setsavoiding distance 1 are of maximal density
2^-n, as conjectured by Bachoc and Robins. We prove that this is true for n =
2, and for the Vorono\"i regions of the lattices An, n >= 2
Combinatorial Space Tiling
The present article studies combinatorial tilings of Euclidean or spherical
spaces by polytopes, serving two main purposes: first, to survey some of the
main developments in combinatorial space tiling; and second, to highlight some
new and some old open problems in this area.Comment: 16 pages; to appear in "Symmetry: Culture and Science
On the Voronoi Conjecture for Combinatorially Voronoi Parallelohedra in Dimension 5
In a recent paper, Garber, Gavrilyuk, and Magazinov [Discrete Comput. Geom., 53 (2015), pp. 245--260] proposed a sufficient combinatorial condition for a parallelohedron to be affinely Voronoi. We show that this condition holds for all 5-dimensional Voronoi parallelohedra. Consequently, the Voronoi conjecture in holds if and only if every 5-dimensional parallelohedron is combinatorially Voronoi. Here, by saying that a parallelohedron is combinatorially Voronoi, we mean that is combinatorially equivalent to a Dirichlet--Voronoi polytope for some lattice , and this combinatorial equivalence is naturally translated into equivalence of the tiling by copies of with the Voronoi tiling of . We also propose a new condition which, if satisfied by a parallelohedron , is sufficient to infer that is affinely Voronoi. The condition is based on the new notion of the Venkov complex associated with a parallelohedron and cohomologies of this complex