The isometries of the cut, metric and hypermetric cones

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 $$\{1,\dots,n\}$$, that is, $$Is(\mathrm{Cut}{n})=Is(\mathrm{Met}{n})\simeq Sym{n}$$ for n ≥ 5. For n = 4 we have $$Is(\mathrm{Cut}{4})=Is(\mathrm{Met}{4})\simeq Sym{3}\times Sym{4}$$. This result can be extended to cones containing the cuts as extreme rays and for which the triangle inequalities are facet-inducing. For instance, $$ Is ({\rm Hyp}_n) \simeq Sym(n)$$ for n ≥ 5, where Hypn denotes the hypermetric cone.