In how many ways a single hyperplane can cut a hypercube? Two "ways" are considered different, if the sets into which they divide vertices of the hypercube are different. So e.g. a line can cut 2-dimensional hypercube in 4 + 2 = 6 ways.

Actually, all I need to know is whether the number of those possible cuts is polynomial or exponential with respect to the number of vertices of the hypercube.

threshold functionsis about twice the number of slicing hyperplanes, so you could look at that literature for a citation, e.g., S. Yajima, T. Ibaraki "A lower bound of the number of threshold functions"IEEE Trans. Comput., EC-14 (1965), pp. 926–929 (which I cannot access). – Joseph O'Rourke Feb 29 '12 at 16:55