An analog of Ordin's theorem for parallelotopes

V. P. Grishukhin

Čebyševskij sbornik, Tome 19 (2018) no. 2, pp. 407-420

Résumé

Parallelotope is a convex polytope in an affine space such that its shifts by vectors of a lattice $L$ fill the space without gaps and intersections by inner points. A special case of a parallelotope is a Dirichlet-Voronoi cell of a lattice with respect to a metric generated by a positive quadratic form. More than 100 years ago G. Voronoi supposed that each parallelotope is a Dirichlet-Voronoi cell of its lattice with respect some metric. A.Ordin introduced notions of an irreducible face and a $k$-irreducible parallelotope whose all faces of codimension $K$ are irreducible. A parallelotope tiling is called $k$-irreducible if its parallelotopes are $k$-irreducible. Ordin proved the conjecture of Voronoi for $3$-irreducible parallelotopes. There are two vectors related to a facet $F$ of a parallelotope. Namely, facet vector $l_F$ of the lattice $L$ of the tiling $\mathcal T$ and normal vector $p_F$ of the facet $F$. The facet vectors integrally generate the lattice $L$. One of the form of Voronoi conjecture asserts that there are such parameters $s(F)$ that scaled (canonical) normal vectors $s(F)p_F$ integrally generate a lattice $\Lambda$. In this paper, uniquely scaled faces are defined. Such a face $G$ determines uniquely up to a multiple parameters $s(F)$ of facets of the tiling $\mathcal T$ containing the face $G$. A tiling whose faces of codimension $k$ are uniquely scaled is $k$-irreducible. It is proved here the following analog of Ordin's Theorem: There exists a canonical scaling of normal vectors of facets of the tiling $\mathcal T$ if, for some integer $k\ge 1$, all its faces of codimension $k$ and $k+1$ are uniquely scaled. The cases $k=2$ and $k=3$ correspond to $2$- and $3$-irreducible tilings of Ordin.

Détail

Keywords: parallelotope, Voronoi conjecture, uniquely scaled normal vectors.