An intermediate value theorem for face polytopes

Аннотация

The paper proves a theorem on polytopal fans and face polytopes that can be treated as an intermediate value theorem for face polytopes. According to this theorem if all fans F_S obtained from a fan F by replacing one of its cones K with a subdivision S of K in some set H are polytopal, then the fan F is polytopal as well. Moreover, if P_S, S ∈ H, are arbitrary face polytopes of the fans F_S, then some positive combination of P_S, S ∈ H, is a face polytope of the fan F. The reverse of the theorem is not true.