Is the Minkowski sum of two faces a face?

Question

Let $K$ and $L$ be two polytopes. Let $F_K$ and $F_L$ be faces of $K$ and $L$ respectively. Is $F_K + F_L$ a faces of $K+L$? Here the sum is the Minkowski sum.

Answer 1

Obviously not. Let $K$ be a square and consider the Minkowski sum $K + K$. The Minkowski sum is a square whose side has twice the length of the square $K$. The Minkowski sum of one side of $K$ and an adjacent side of $K$ is a square with the same size as $K$, so it's not a face of $K + K$.

Answer 2

The answer is no. As the example already given explains it quite well. However, if you have a linear functional $f$ and you are computing the face of $K + L $ that maximizes the functional $f$, then you have a quite easy relation

$$ (K + L)_f = K_f + L_f $$

Here I use $K_f$ to denote the face of $K$ that maximizes the functional $f$.

This also shows that if you have two faces $F_K$ and $F_L$ of the polytopes, then it's sum is a face of $K+L$ if they maximize the same functional. This is in fact an if and only if statement!