Find all hyperplanes separating unique sets of `k` points
If I understand well, you are interested in the $k$-set problem:
http://en.wikipedia.org/wiki/K-set_(geometry)

In general their number can be $O(n^{\lfloor d/2\rfloor}k^{\lceil d/2\rceil})$ and though I could not find references now, I think you can also compute them in about this much time.