Closure under Minkowski sum. The Minkowski sum of two sets of vectors $A, B \in R^d$ is given by 

$$ A \oplus B  = \{ a + b \mid a \in A, b \in B \}$$

I just heard an interesting problem (attributed to Dan Halperin): Given a shape $B$, does there exist a shape $A$ such that $A \oplus A = B$ ?

But that's not my question (it appears to be an open problem). Observe that in the above problem, if $B$ is a convex set, then there exists a solution $A = (1/2)B$ since convex sets are closed under the taking of Minkowski sums. 

Fix a class of shapes ${\cal S}$. We say that ${\cal S}$ is closed under Minkowski sums if for any $A, B \in {\cal S}, A \oplus B \in {\cal S}$. 

So my question is:


  Is there a nice characterization of
  classes of shapes ${\cal S}$ that are
  closed under Minkowski sums ?
Lattices and linear subspaces are closed under Minkowski sum. Thats more or less immediate from their definition. Lattices+linear subspaces are closed under Minkowski sum (i.e., a member of this set is for example a set of parallel lines in distance 1 from each other). Connected polygons with holes are closed under Minkowski sum. Rings [the set differences of two concentric disks] are closed under Minkowski sum (a disk is considered to be a ring, naturally). The set of line segments parallel to a certain direction are closed under Minkowski sum. Mushed potatoes are closed under Minkowski sum, but only if they are well cooked (or maybe not, it is too late)...

Also, the family of finite union of concentric rings is closed under Minkowski sum.