Let $A$ and $B$ be polygons in the plane, each with $n$ vertices. Their Minkowski sum is defined as: $$A + B = \{a + b \mid a \in A, b \in B \}$$ and has the property: $$ |\text{Conv}(A+B)| \leq 2n $$ where $|...| $ denotes the number of vertices in a set.
If we define a partial Minkowski sum as: $$A \otimes B = \{a + b \mid (a, b) \in C, C \subset A \times B \}$$ does the following property hold? $$ |\text{Conv}(A \otimes B)| \leq 2n $$
I'm having difficulty adapting the standard proof (in e.g. De Berg's Computational Geometry), as the partial sum can introduce new edges into $\text{Conv}(A \otimes B)$ which exist in neither $A$ nor $B$. Ideally, I would like to generalize the result to multiple polygons.
In a related question https://mathoverflow.net/questions/27040/a-variation-of-minkowski-sum this operation is dubbed a 'sumset along a graph' rather than a partial Minkowski sum.