Let $A,B \subset \mathbb R^n$ be compact measurable subsets and let $A + B = \{ a+b \mid a \in A, b \in B \}$. If $m$ is the Lebesgue measure, is it possible to relate $m(A + B)$ to $m(A)$ and $m(B)$?
In $\mathbb R$, for instance, $m([a,b] + [c,d]) = m([a+c, b+d]) = (b-a) + (d-c) = m([a,b]) + m([c,d])$. I would be happy, in general, with something like $m(A+B) \le m(A) + m(B)$ but I do not know whether this is possible.