TITLE: Proof: Minkowski sum polytope implies A and B polytopes
QUESTION [5 upvotes]: Suppose $A$ and $B$ are convex sets and their Minkowski sum $A+B$ is a polytope. 
How do you prove that $A$ and $B$ are polytopes as well?

REPLY [3 votes]: HINT:
$A+B$ being a polytope, it is the convex of finitely many points $a_i + b_i$, $1\le i \le n$, where $a_i \in A$ and $b_i \in B$. Let $\tilde A = \text{Co}( a_i)$, $\tilde B = \text{Co}(b_i)$. One checks right away that $\tilde A + \tilde B \supset A+B$ and therefore we have the equality 
$$\tilde A + \tilde B = A+B$$
while $\tilde A \subset A$ and $\tilde B \subset B$.  From here one shows that $\tilde A = A$ and $\tilde B = B$ ( reduce to the case of $\dim =1$).