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The balanced minimal evolution (BME) method of creating phylogenetic trees can be formulated as a linear programming problem, minimizing an inner product over the vertices of the BME polytope. In this paper we undertake the project of describing the facets of this polytope. We classify and identify the combinatorial structure and geometry (facet inequalities) of all the facets in dimensions up to five, and classify even more facets in all dimensions. A full set of facet inequalities would allow a full implementation of the simplex method for finding the BME tree-although there are reasons to think this an unreachable goal. However, our results provide the crucial first steps for a more likely-to-be-successful program: finding efficient relaxations of the BME polytope.

Facets of the balanced minimal evolution polytope