The number of partitions of a set of N points in k dimensions induced by hyperplanes

An arbitrary ( k – 1)-dimensional hyperplane disconnects K -dimensional Euclidean space E k into two disjoint half-spaces. If a set of N points in general position in E k is given [no k +1 in a ( k –1)-plane, no k in a ( k –2)-plane, and so on], then the set is partitione into two subsets by the hyperplane, a point belonging to one or the other subset according to which half-space it belongs to; for this purpose the half-spaces are considered as an unordered pair.