594 research outputs found
The decomposition of the hypermetric cone into L-domains
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 are key for the decomposition of
\HYP_5.Comment: 20 pages 2 figures, 2 table
Morphisms and order ideals of toric posets
Toric posets are cyclic analogues of finite posets. They can be viewed
combinatorially as equivalence classes of acyclic orientations generated by
converting sources into sinks, or geometrically as chambers of toric graphic
hyperplane arrangements. In this paper we study toric intervals, morphisms, and
order ideals, and we provide a connection to cyclic reducibility and conjugacy
in Coxeter groups.Comment: 28 pages, 8 figures. A 12-page "extended abstract" version appears as
[v2
The isometries of the cut, metric and hypermetric cones
We show that the symmetry groups of the cut cone Cut(n) and the metric cone
Met(n) both consist of the isometries induced by the permutations on {1,...,n};
that is, Is(Cut(n))=Is(Met(n))=Sym(n) for n>4. For n=4 we have
Is(Cut(4))=Is(Met(4))=Sym(3)xSym(4).
This is then extended to cones containing the cuts as extreme rays and for
which the triangle inequalities are facet-inducing. For instance,
Is(Hyp(n))=Sym(n) for n>4, where Hyp(n) denotes the hypermetric cone.Comment: 8 pages, LaTeX, 2 postscript figure
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