  The hypermetric cone $\HYP_{n+1}$ is the parameter space of basic Delaunay
polytopes in n-dimensional lattice. The cone $\HYP_{n+1}$ is polyhedral; one
way of seeing this is that modulo image by the covariance map $\HYP_{n+1}$ is a
finite union of L-domains, i.e., of parameter space of full Delaunay
tessellations.
  In this paper, we study this partition of the hypermetric cone into
L-domains. In particular, it is proved that the cone $\HYP_{n+1}$ of
hypermetrics on n+1 points contains exactly {1/2}n! principal L-domains. We
give a detailed description of the decomposition of $\HYP_{n+1}$ for n=2,3,4
and a computer result for n=5 (see Table \ref{TableDataHYPn}). Remarkable
properties of the root system $\mathsf{D}_4$ are key for the decomposition of
$\HYP_5$.
