Random Translates in Minkowski Sums

Suppose that $A$ and $B$ are sets in ${\mathbb{R}}^d$, and we form the sumset of $A$ with $n$ random points of $B$. Given the volumes of $A$ and $B$, how should we choose them to minimize the expected volume of this sumset? Our aim in this paper is to show that we should take $A$ and $B$ to be Euclidean balls. We also consider the analogous question in the torus ${\mathbb{T}}^d$, and we show that in this case the optimal choices of $A$ and $B$ are bands, in other words, sets of the form $[x,y]\times {\mathbb{T}}^{d-1}$. We also give stability versions of our results.