# The isometries of the cut, metric and hypermetric cones

Research paper by Antoine Deza, Boris Goldengorin, Dmitrii V. Pasechnik

Indexed on: 02 Jun '03Published on: 02 Jun '03Published in: Mathematics - Metric Geometry

#### 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.