I was wondering whether the following statement about a Minkowski sum of two polytopes is true:
Let $P=P_1 + P_2$ be the Minkowski sum of two polytopes $P_1$ and $P_2$ in $\Bbb R^n$, i.e. $P=\{p_1+p_2: p_1\in P_1, p_2 \in P_2\}$. Then the number of edges (1-dim. faces) of $P$ is at least the maximum of the number of edges of $P_1$ and $P_2$, i.e. if $f_1(P)$ denotes the number of edges of $P$, then $f_1(P) \geq \max\{f_1(P_1),f_1(P_2)\}$. For my purposes, one may assume that the origin lies in the interior of $P_1$ and $P_2$.
Can someone please provide a reference, proof, or counterexample? (Or state if it's unknown). Thank you in advance.
As a follow-up question: does $P$ have more vertices than $P_1$ and $P_2$? What about facets, or more generally, $k$-faces?
First you have to consider the extremal case of 2 mutually perpendicular polytopes $P$ and $Q$. There the Minkowski sum then simply is just
$$(P,0) + (0,Q) = P \times Q$$
the according Cartesian product or duoprism. For that one you clearly would have
$$f_1(P \times Q) = f_0(P) f_1(Q) + f_1(P) f_0(Q)$$
Next you would have to consider quite generally (any relation of $P$ and $Q$) that $P+Q$ is describable as some projection or shadow of $P \times Q$.
--- rk