engleski

The decomposition of the hypermetric cone into L-domains

The hypermetric cone HYP(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 HYP(n+1) is the union of a finite set 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, we prove that the cone HYP(n+1) of hypermetrics on n+1 points contains exactly n!/2 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. Remarkable properties of the root system D4 are key for the decomposition of HYP(5).

Lattice ; L-type ; matroid

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