Convex polyhedra of binary trees and their symmetries

Problem of finding minimal parametric fillings of a finite metric space $M$ can be reduced to classical linear programming. The set of admissible values of the dual problem is a convex polyhedron $\Lambda_G$ that depends on the filling type $G$, i.e., on a tree, whose set of degree $1$ vertices equals $M$, and all other vertices have degree $3$ (such trees are referred as binary trees connecting $M$). Vertices of this polyhedron have an important role in minimal filling weight calculation.
Generally speaking, isomorphic binary trees connecting the same space $M$ correspond to different polyhedra. In the present paper a complete answer on the relations between such polyhedra is obtained. Besides, the question on the structure of the set $U$ obtained as the union of the vertices of all polyhedra $\Lambda_G$ over all binary trees connecting a given set $M$. This set $U=U(m)$ depends on the number of points $m$ in the set $M$. It turns out that the set $U(m)$ is convex for $m\le6$, i.e., it is a vertex set of a convex polyhedron. Convexity of $U(m)$ for other $m$ is an open question.