Is there a characterization of measure zero subsets $A$ of $\mathbb R^n$, $n>1$  such that the set $A+A$ contains interior? Here $A+A$ is the set of points $\{ x+y \mid x, y\in A \}$. 

Is it true that if the convex hull of the connected component of $A$ contains interior then so does $A+A$?