Cross-posted from https://math.stackexchange.com/questions/65195/minimum-cardinality-of-a-difference-set-in-mathbb-rn.

Given a finite set $S$ of $m$ points in $\mathbb R^n$ that do not all lie in the same $(n-1)$-dimensional hyperplane, consider the set of difference vectors:

$\{x-y \, | \, x,y \in S\}$

What is the minimum cardinality of this set, as a function of $m$ and $n$?

(The sets that minimize this should be "small" subsets of a lattice, but I don't know what specific shapes minimize it.)

What is the status of exact results for this problem for small $n$ (say $n = 2$ or $3$)?