Affine maps that induce polyhedral complex isomorphisms (Q1577547)

An affine map between two polyhedral complexes which is bijective on the underlying sets need not induce an isomorphism of the complexes preserving the induced cell-complex structure. The key observation in this paper is that this is nevertheless true under an additional assumption: If an affine bijection \(f: T_1\to T_2\) between two polyhedral complexes \(T_1, T_2\), both of which consist of a union of faces of two convex polyhedra \(P_1\), \(P_2\), extends to an affine map from \(P_1\) to \(P_2\) then \(f\) necessarily respects the cell-complex structure of \(T_1\) and \(T_2\) induced by \(P_1\) and \(P_2\), respectively. This result depends on certain properties of extremal points and extremal subsets of polyhedra. There are applications in the sciences, including biology and molecular evolution.