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. I think this falls into the realm of "additive combinatorics" or "arithmetic combinatorics", but there aren't tags for those.)