Affine maps that induce polyhedral complex isomorphisms

In this paper we show that an affine bijection f: T-1 --> T-2 between two p olyhedral complexes T-1, T-2, both of which consist of a union of faces of two convex polyhedra P-1 and P-2, necessarily respects the cell-complex str ucture of T-1 and T-2 inherited from P-1 and P-2, respectively, provided f extends to an affine map from P-1 into P-2. In addition, we present an appl ication of this result within the area of T-theory to obtain a far-reaching generalization of previous results regarding the equivalence of two distin ct constructions of the phylogenetic tree associated to "perfect" (that is, treelike) distance data.