Alexey Garber

The complete classification of five-dimensional Dirichlet–Voronoi polyhedra of translational lattices

This paper reports on the full classification of Dirichlet-Voronoi polyhedra and Delaunay subdivisions of five-dimensional translational lattices. A complete list is obtained of 110 244 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, 181 394 contraction types are obtained. The paper gives details of the computer-assisted enumeration, which was verified by three independent implementations and a topological mass formula check.

Belt distance between facets of space-filling zonotopes

To every d-dimensional polytope P with centrally symmetric facets one can assign a “subway map” such that every line of this “subway” contains exactly the facets parallel to one of the ridges of P. The belt diameter of P is the maximum number of subway lines one needs to use to get from one facet to another. We prove that the belt diameter of a d-dimensional space-filling zonotope does not exceed ⌈log2(4/5)d⌉.

On \({\pi}\)-Surfaces of Four-Dimensional Parallelohedra

We show that every four-dimensional parallelohedron P satisfies a recently found condition of Garber, Gavrilyuk & Magazinov sufficient for the Voronoi conjecture being true for P. Namely, we show that for every four-dimensional parallelohedron P the one-dimensional homology group of its

On a Helly-type question for central symmetry

{\pi}

Another Ham Sandwich in the Plane

π-surface is generated by half-belt cycles.

Graphs of linear operators

A @P-zonotope is a zonotope that can be obtained from permutahedron by deleting zone vectors. Any face F of codimension 2 of such zonotope generates its belt, i.e. the set of all facets parallel to F. The belt diameter of a given zonotope Z is the diameter of the graph with vertices correspondent to pairs of opposite facets and with edges connect facets in one belt. In this paper we investigate belt diameters of @P-zonotopes. We prove that any d-dimensional @P-zonotope (d>=3) has belt diameter at most 3. Moreover if d is not greater than 6 then its belt diameter is bounded from above by 2. Also we show that these bounds are sharp. As a consequence we show that diameter of the edge graph of dual polytope for such zonotopes is not greater than 4 and 3 respectively.

The Voronoi Conjecture for Parallelohedra with Simply Connected δ-Surfaces

We study a certain Helly-type question by Konrad Swanepoel. Assume that X is a set of points such that every k-subset of X is in centrally symmetric convex position, is it true that X must also be in centrally symmetric convex position? It is easy to see that this is false if