Invariance of Minkowski sum of sets

Question

Given an euclidean space $E$, two sets $A,B\subset E$ and the action on $E$ of two groups $G_A,G_B$ such that $G_A A=A$ and $G_B B=B$, it is possible to generate a group that leaves invariant $A\oplus B$, the Minkowski sum of $A$ and $B$, from the groups $G_A$ and $G_B$?

Answer 1

Suppose $A$ and $B$ are compact, then we can assume both groups $G_A$ and $G_B$ fix the origin. In this case, $A+B$ is invariant for $G_A\cap G_B$.

It is easy to construct example when $G_A\cap G_B$ includes all the symmetries of $A+B$, but in general this group might be much larger. For example, a round disc is a sum of two Reuleaux triangles.