arXiv subjects

Alexey Garber

Publications and source records attributed to Alexey Garber.

32 records · Page 2Linked to original sources

arXiv2018

On the Voronoi Conjecture for combinatorially Voronoi parallelohedra in dimension five

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 $\mathbb R^5$ holds if and only if every five-dimensional parallelohedron is combinatorially Voronoi. Here, by saying that a parallelohedron $P$ is combinatorially Voronoi, we mean that the tiling $\mathcal T(P)$ by translates of $P$ is combinatorially isomorphic to some tiling $\mathcal T(P')$, where $P'$ is a Voronoi parallelohedron, and that the isomorphism naturally induces a linear isomorphism of lattices $Λ(P)$ and $Λ(P')$. 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.

math.MG↗

arXiv2018

Periodic triangulations of $\mathbb{Z}^n$

We consider in this work triangulations of $\mathbb{Z}^n$ that are periodic along $\mathbb{Z}^n$. They generalize the triangulations obtained from Delaunay tessellations of lattices. Other important property is the regularity and central-symmetry property of triangulations. Full enumeration for dimension at most $4$ is obtained. In dimension $5$ several new phenomena happen: there are centrally-symmetric triangulations that are not Delaunay, there are non-regular triangulations (it could happen in dimension $4$) and a given simplex has a priori an infinity of possible adjacent simplices. We found $950$ periodic triangulations in dimension $5$ but finiteness is unknown.

math.CO↗

arXiv2018

On the Origin of Crystallinity: a Lower Bound for the Regularity Radius of Delone Sets

The local theory of regular or multi-regular systems aims at finding sufficient local conditions for a Delone set $X$ to be a regular or multi-regular system. One of the main goals is to estimate the regularity radius $\hatρ_d$ for Delone sets $X$ in terms of the radius $R$ of the largest "empty ball" for $X$. The present paper establishes the lower bound $\hat{ρ_d}\geq 2dR$ for all $d$, which is linear in $d$. The best previously known lower bound had been $\hatρ_d\geq 4R$ for $d\geq 2$. The proof of the new lower bound is accomplished through explicit constructions of Delone sets with mutually equivalent $(2dR-\varepsilon)$-clusters, which are not regular systems.

math.MG↗

arXiv2018

Weighted $1\times1$ cut-and-project sets in bounded distance to a lattice

Recent results of Grepstad and Lev are used to show that weighted cut-and-project sets with one-dimensional physical space and one-dimensional internal space are bounded distance equivalent to some lattice if the weight function $h$ is continuous on the internal space, and if $h$ is either piecewise linear, or twice differentiable with bounded curvature.

math.MG↗

arXiv2018

Pisot substitution sequences, one dimensional cut-and-project sets and bounded remainder sets with fractal boundary

This paper uses a connection between bounded remainder sets in $\mathbb{R}^d$ and cut-and-project sets in $\mathbb{R}$ together with the fact that each one-dimensional Pisot substitution sequence is bounded distance equivalent to some lattice in order to construct several bounded remainder sets with fractal boundary. Moreover it is shown that there are cut-and-project sets being not bounded distance equivalent to each other even if they are locally indistinguishable, more precisely: even if they are contained in the same hull.

math.MG↗

arXiv2018

On a Helly-type question for central symmetry

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 $k\le 5$, but it may be true for sufficiently large $k$. We investigate this question and give some partial results.

math.MG↗

arXiv2016

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 $110244$ 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 $181394$ contraction types. We report on details of our computer assisted enumeration, which we verified by three independent implementations and a topological mass formula check.

math.MG↗

arXiv2014

The Voronoi conjecture for parallelohedra with simply connected $δ$-surface

We show that the Voronoi conjecture is true for parallelohedra with simply connected $δ$-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 $π$-surface associated with a parallelohedron and give another condition in terms of homology group of the constructed surface. Every parallelohedron with simply connected $δ$-surface also satisfies the condition on homology group of the $π$-surface.

math.MG↗

arXiv2014

Symmetries of Monocoronal Tilings

The vertex corona of a vertex of some tiling is the vertex together with the adjacent tiles. A tiling where all vertex coronae are congruent is called monocoronal. We provide a classification of monocoronal tilings in the Euclidean plane and derive a list of all possible symmetry groups of monocoronal tilings. In particular, any monocoronal tiling with respect to direct congruence is crystallographic, whereas any monocoronal tiling with respect to congruence (reflections allowed) is either crystallographic or it has a one-dimensional translation group. Furthermore, bounds on the number of the dimensions of the translation group of monocoronal tilings in higher dimensional Euclidean space are obtained.

math.MG↗

arXiv2013

Another ham sandwich in the plane

We show that every two nice measures in the plane can be partitioned into equal halves by translation of an angle from arbitrary k-fan when k is odd and in some cases when k is even. We also give some counterexamples for certain fans and measures.

math.MG↗

arXiv2013

On π-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 group of rational first homologies of its π-surface is generated by half-belt cycles.

math.MG↗

arXiv2012

Belt diameter of $Π$-zonotopes

We prove that any d-dimensional zonotope obtained from permutahedron by deleting zone vectors 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 then 4 and 3 respectively.

math.CO↗

arXiv2011

The second Voronoi conjecture on parallelohedra for zonotopes

We prove the second Voronoi conjecture on parallelohedra for zonotope. We show that for a given face-to-face tiling of d-dimensional Euclidean space into parallel copies of zonotope Z there are d vectors, connecting centers of zonotopes with common facet, that are basis of the correspondent lattice of the tiling.

math.CO↗

arXiv2010

Belt distance between facets of space-filling zonotopes

For every d-dimensional polytope P with centrally symmetric facets we can associate a "subway map" such that every line of this "subway" corresponds to set of facets parallel to one of ridges P. The belt diameter of P is the maximal number of line changes that you need to do in order to get from one station to another. In this paper we prove that belt diameter of d-dimensional space-filling zonotope is not greater than $\lceil \log_2\frac45d\rceil$. Moreover we show that this bound can not be improved in dimensions d at most 6.

math.CO↗