On Convex Sets Associated with Permutations, Path-metrics, and Cuts

In a recent preprint by Amaral & Letchford (2006) convex hulls of sets of matrices corresponding to permutations and path-metrics are studied. A symmetric n × n-matrix is a path metric, if there exist points x1, . . . , xn ∈ R such that the matrix entries are just the pairwise distances |xk − xl| between the points and if these distances are at least one whenever k 6= l. The convex hull of the path metrics is denoted by Qn. The convex hull of the subset of matrices for which there exists a permutation π such that xk = π(k) for all k is denoted by Pn. Pn is a polytope. Amaral & Letchford show that Pn is an exposed subset of Qn, and that the closure of Qn is a polyhedron, namely, the Minkowski sum of Pn and the cut-cone. Amaral & Letchford leave open the question of whether Qn is closed or not. In this paper we first give a structural theorem relating Qn to the permutahedron. We show that it is the convex hull of n!/2 pairwise disjoint simplicial cones of dimension n − 1. We use this result to characterize, for each extreme point X of Qn, all unbounded one-dimensional extreme subsets of Qn containing X. Second, we characterize the set of unbounded edges of the closure of Qn ending at each vertex. This result shows a relationship with the permutahedron, too. As a by-product of these two theorems, we obtain that, for n ≥ 4, the convex set Qn is not closed.