  Balanced minimum evolution is a distance-based criterion for the
reconstruction of phylogenetic trees. Several algorithms exist to find the
optimal tree with respect to this criterion. One approach is to minimize a
certain linear functional over an appropriate polytope. Here we present
polytopes that allow a similar linear programming approach to finding
phylogenetic networks. We investigate a two-parameter family of polytopes that
arise from phylogenetic networks, and which specialize to the Balanced Minimum
Evolution polytopes as well as the Symmetric Travelling Salesman polytopes. We
show that the vertices correspond to certain level-1 phylogenetic networks, and
that there are facets or faces for every split. We also describe minimal facets
and a family of faces for every dimension.
