Proof: Minkowski sum polytope implies A and B polytopes

Question

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?

Answer 1

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 $\dim =1$).