On Page 8 (or Page 250 as mentioned in the article) of BASIC PROPERTIES OF CONVEX POLYTOPES, under the definition of Minkowski sum it is given that $P+\lambda P'$ is combinatorially equivalent for all $\lambda>0.$
Two polytopes are combinatorially equivalent if their face lattices are the same. So, for example, I was trying to prove that $P+P'$ and $P+2P'$ are combinatorially equivalent by showing that there is inclusion-preserving one-to-one correspondence $\phi$ between their faces. If $F$ is a face of $P+P'$ then $F=F_1+F_2$ for some $F_1\in P$ and $F_2\in P'$. But I am unsure what $\phi(F)$ would be in this case as Minkowski sum of two faces need not be a face.
Any hint or direction?