Math 121c: Topics in Geometric Combinatorics, Spring 2012 Problems-
(a) Let P, Q ⊂ Rn be two n-dimensional polytopes. We say P and Q are combinatorially equivalent if there is a bijection φ: F(P) → F(Q) that is order preserving. That is, if F1, F2 ∈ F(P), then F1 ⊂ F2 if and only if φ(F1) ⊂ φ(F2).
We say that P and Q are affinely equivalent if there is an affine linear transformation φ: Rn → Rn (i.e. a linear transformation together with a translation) that maps P to Q bijectively.
Prove that if P, Q are affinely equivalent then they are combinatorially equivalent. Is the converse true?
(b) Prove that P ⊂ Rn is an H-polytope if and only if P is a V-polytope. (Hint: For the forward direction, use induction on n. Then, prove the backward direction by exploiting duality.)