Once more about Voronoi's conjecture and space tiling zonotopes

Abstract

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 $\gamma_i$ and a positive definite $n\times n$ matrix $Q$ such that $\gamma_i p_i=Qt_i$ 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 $Q=(Z_{\beta}Z^T_{\beta})^{-1}$ 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 $Z_{\beta}$ are the vectors $\sqrt{2\beta_j}z_j$, where the scalars $\beta_j$, $1\le j\le r$, are such that the system of vectors $\{\beta_jz_j:1\le j\le r\}$ is unimodular. $P(Z)$ defines a dicing lattice which is the set of intersection points of the dicing family of hyperplanes $H(j,k)=\{x:x^T(\beta_jQz_j)=k\}$, where $k$ takes all integer values and $1\le j\le r$.