On the Dirichlet-Voronoi cell of unimodular lattices
Let \(L\) be an \(n\)-dimensional integral, unimodular lattice and \(x\) a vector of \(L\). Then \(L_x:=L\cap \text{aff} L_x\) is a sublattice of dimension \(n-1\) orthogonal to \(x\). Denote by \(H_x\) the subspace generated by \(L_x\). The lattice vectors corresponding to the hyperplanes which contain a proper facet of the Dirichlet-Voronoi (D-V) cell \(D\) of \(L\) are called relevants. The height \(k_x\), \(x\in L\), is the number of lattice hyperplanes parallel to and different from \(H_x\) intersecting the segment \([0,x]\). Let \(e_x\) be a lattice vector such that \(L= \cup \{(L_x+i \cdot e_x) \mid i\in \mathbb{Z}\}\). Then the lattice sections of \(D\) can be written as \(D\cap (H_x+i \cdot e_x)\) for \(i\in \left\{-\left[{k_x-1 \over 2} \right], \left[{k_x \over 2} \right] \right\}\). The first part of the paper gives relationships between the volumes of lattice sections and the volume of the D-V cell \(D_x\) of \(L_x\); for example \[ \text{vol} (D_x)= \sum^{\left[{k_x \over 2} \right]}_{i= -\left[{k_x-1 \over 2} \right]} \text{vol} \bigl(D \cap(H_x+ i\cdot e_x) \bigr) \] and \[ \text{vol} (D)= {|x|\over k_x} \sum^{\left[{k_x \over 2} \right]}_{i=-\left[{k_x-1 \over 2} \right]} \text{vol} \bigl(D \cap(H_x+i \cdot e_x) \bigr)= |x|M_x, \] where \(M_x\) is the average value of the volumes of the lattice sections of \(D\) orthogonal to the direction \(x\). The remainder of the paper is concerned with a generalization of a result of \textit{G. F. Voronoi} [J. Reine Angew. Math. 136, 67-178 (1909; JFM 40.0267.17)], which says that the lattice vector \(x\) is a relevant of the cell \(D\) iff it is a minimal vector of the coset \(x+ 2L\), and, if \(y\) is any other minimal element of this coset, then \(y=\pm x\).
scientific article; zbMATH DE number 942748