The decomposition of the hypermetric cone into LL-domains

The hypermetric cone HY Pn+1HY Pn+1 is the parameter space of basic Delaunay polytopes of nn-dimensional lattice. If one fixes one Delaunay polytope of the lattice then there are only a finite number of possibilities for the full Delaunay tessellations. So, the cone HY Pn+1HY Pn+1 is the union of a finite set of LL-domains, i.e. of parameter space of full Delaunay tessellations.In this paper, we study this partition of the hypermetric cone into LL-domains. In particular, we prove that the cone HY Pn+1HY Pn+1 of hypermetrics on n+1n+1 points contains exactly 12n! principal LL-domains. We give a detailed description of the decomposition of HY Pn+1HY Pn+1 for n=2,3,4n=2,3,4 and a computer result for n=5n=5. Remarkable properties of the root system D4 are key for the decomposition of HY P5HY P5.