The isometries of the cut, metric and hypermetric cones - Deza, Antoine. (2011)
We show that the symmetry groups of the cut cone Cutn and the metric cone Metn both consist of the isometries induced by the permutations on {1, . . . , n}; that is, Is(Cutn) = Is(Metn) ≃ Sym(n) for n ≥ 5. For n = 4 we have Is(Cut4) = Is(Met4) ≃ Sym(3) × Sym(4). This result can be extended to cones containing the cuts as extreme rays and for which the triangle inequalities are facet-inducing. For instance, Is(Hypn) ≃ Sym(n) for n ≥ 5, where Hypn denotes the hypermetric cone.