Find all hyperplanes separating unique sets of `k` points I have a series of n points in d-dimensional continuous space.  I want to find a series of hyperplanes such that k points are below the hyperplane and that the set of k points below each hyperplane is a unique set.  It does not matter what the hyperplane is (e.g. if it connects d points or if it is another hyperplane not intersecting any points).  I can discretise the space if it helps.  "below" is defined as the negative direction for the first dimension.

The way I have approached is to try all hyperplanes defined by all sets of d points, project each point on to this hyperplane, and test if the projection is above the point.  This results in n^(d+1) complexity.

Is there a better way?  What is it?
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.