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The isometries of the cut, metric and hypermetric cones

Antoine Deza·Boris Goldengorin·Dmitrii V. Pasechnik
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Abstract

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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Antoine Deza, Boris Goldengorin, Dmitrii V. Pasechnik. 2003-06-02. The isometries of the cut, metric and hypermetric cones. https://doi.org/10.1007/s10801-006-6924-6

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