On zone-reduction in Voronoi-cells of lattices of the first kind

The classification of Voronoi--cells (up to combinatorial equivalence) of lattices of the first kind in a fixed dimension is realized by successive zone--reductions in the permutahedron $\Pi_n$ which is itself the Voronoi--cell of the dual root lattice $\mathsf{A}^*_n$. The zones of such a Voronoi--cell are in one--to--one correspondence with the Selling--parameters of an obtuse superbase of the lattice belonging to this cell. The fact that only those zones of the permutahedron are reduced to points which correspond to the non--vanishing Selling--parameters follows from a formula for calculating the zone lengths which has been conjectured by {\sc R.\ Scharlau} and is proved here.

Preprint, January 2000.   (dvi-file, 29KB)