Once more about Voronoi's conjecture and space tiling zonotopes
| dc.creator | Deza, Michel | |
| dc.creator | Grishukhin, Viacheslav | |
| dc.date | 2002-03-13 | |
| dc.date.accessioned | 2026-07-07T04:47:01Z | |
| dc.date.available | 2026-07-07T04:47:01Z | |
| dc.description | Voronoi conjectured that any parallelotope is affinely equivalent to a Voronoi polytope. A parallelotope is defined by a set of $m$ facet vectors $p_i$ and defines a set of $m$ lattice vectors $t_i$, $1\le i\le m$. We show that Voronoi's conjecture is true for an $n$-dimensional parallelotope $P$ if and only if there exist scalars $γ_i$ and a positive definite $n\times n$ matrix $Q$ such that $γ_i p_i=Qt_i$ for all $i$. In this case the quadratic form $f(x)=x^TQx$ is the metric form of $P$. As an example, we consider in detail the case of a zonotopal parallelotope. We show that $Q=(Z_βZ^T_β)^{-1}$ for a zonotopal parallelotope $P(Z)$ which is the Minkowski sum of column vectors $z_j$ of the $n\times r$ matrix $Z$. Columns of the matrix $Z_β$ are the vectors $\sqrt{2β_j}z_j$, where the scalars $β_j$, $1\le j\le r$, are such that the system of vectors $\{β_jz_j:1\le j\le r\}$ is unimodular. $P(Z)$ defines a dicing lattice which is the set of intersection points of the dicing family of hyperplanes $H(j,k)=\{x:x^T(β_jQz_j)=k\}$, where $k$ takes all integer values and $1\le j\le r$. | |
| dc.description | 10 pages, no figures | |
| dc.identifier | https://arxiv.org/abs/math/0203124 | |
| dc.identifier | http://arxiv.org/abs/math/0203124 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/63554 | |
| dc.subject | Metric Geometry | |
| dc.subject | Combinatorics | |
| dc.subject | Primary 52B22, 52B12; Secondary 05B35, 05B45 | |
| dc.title | Once more about Voronoi's conjecture and space tiling zonotopes | |
| dc.type | text |