The decomposition of the hypermetric cone into -domains

The hypermetric cone HY P n + 1 is the parameter space of basic Delaunay polytopes of n -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 P n + 1 is the union of a finite set of L -domains, i.e. of parameter space of full Delaunay tessellations. s paper, we study this partition of the hypermetric cone into L -domains. In particular, we prove that the cone HY P n + 1 of hypermetrics on n + 1 points contains exactly 1 2 n ! principal L -domains. We give a detailed description of the decomposition of HY P n + 1 for n = 2 , 3 , 4 and a computer result for n = 5 . Remarkable properties of the root system D 4 are key for the decomposition of HY P 5 .