I'd like to show (or disprove) the claim that the Minkowski sum of two triangles with corners in $\mathbb{R}^3$, $A+B$, is equal to the union of the unions of the Minkowski sums of $A$ along all edges of $B$ and the $B$ along all edges of $A$:

 $
A + B = \cup \left[ \cup_{e \in A}\ A + e\ ,\  \cup_{e \in B}\ B + e \right] =: C
 $

where $e \in A$ denotes an edge-segment $e$ on the boundary of $A$.

I believe it's enough to show that $C$ is convex.