Intersection sets in AG(n,q) and a characterization of the hyperbolic quadric in PG(3,q)

Bruen proved that if A is a set of points in AG(n, q) which intersects every hyperplane in at least t points, then |A| ≥ (n+t-1)(q-1) + 1, leaving as an open question how good such bound is. Here we prove that, up to a trivial case, if t > ((n - 1)(q - 1) + 1)/2, then Bruen's bound can be improved. If t is equal to the integer part of ((n - 1)(q - 1) + 1)/2, then there are some examples which attain such a lower bound. Somehow, this suggests the following combinatorial characterization: if a set S of points in PG(3, q) meets every affine plane in at least q - 1 points and is of minimum size with respect to this property, then S is a hyperbolic quadric.