The isometries of the cut, metric and hypermetric cones

Details The isometries of the cut, metric and hypermetric cones The isometries of the cut, metric and hypermetric cones

2003-06-02

arxiv.org/abs/math/0306049v1

J. Algebraic Combinatorics, 23(2006), 197--203

Mathematics

Metric Geometry

8 pages, LaTeX, 2 postscript figures

Scientific paper

10.1007/s10801-006-6924-6

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.

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