arXiv subjects

Andrey Gavrilyuk

Publications and source records attributed to Andrey Gavrilyuk.

2 recordsLinked to original sources

arXiv2015

Geometry of lifts of tilings of euclidean spaces

This paper provides explicit justification for a method of canonical scalings of tilings of euclidean spaces. We present a new combinatorially-geometrical approach for constructing a generatriss of a tiling. The approach is based on an operation of lifting of a tile up to a lifted neighbour. We use this approach and give a new short geometrical proof of a fundamental theorem of theory of parallelotopes: Voronoi's Conjecture holds for a given parallelotope $P$ if and only if the corresponding tiling $\mathcal T_P$ admits a canonical scaling.

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↗