On partitions of N points

Abstract

In a paper (1) by Harding there is a tacit invitation to seek the connection between the following two problems:(i) Find the number, t] k (N), of regions into which a Ar-dimensional space is partitioned by a set of N (k-l)-dimensional hyperplanes. (ii) Find the number, v k (N), of distinct partitions of a given set of N points in a ^-dimensional space E that can be induced by (k-l)-dimensional hyperplanes. Schlafli (2) solved the first problem and Harding (1) solved the second. I wish to show that the first problem can be expressed as a dual of the second and thus provide an alternative derivation of Harding's result.First, form the dual E, of E in the following way. Let the dual of a point u in E be the half-space V = {r | u . r ^ 1} of E; in particular let the dual of the origin of E be all of E. Let the dual of the half-space V={r\v.r£l) of E be the point v of E. Note that u 6 V^>v . u ^ l=>y e U, so incidence properties of points and half-spaces are preserved in the transformation. Now consider N points in E and place the origin at one of the points. The dual of the N points will be (JV-1) half-spaces and the whole space of E, dividing E into r\ k {N-1) regions, each expressible as an intersection I of some of the half-spaces and the complements of the remaining half-spaces.Let a (k -l)-dimensional hyperplane in E separate the N points into two parts-a set P containing the origin and another set P'. Let 8P denote the family of all half-spaces that contain P and are disjoint with i". The dual of 9 is a set 2P of points E; and each point of (P will belong to the intersection Ip of the half-spaces which are duals of points of P and the complements of the half-spaces which are duals of points of P'. Conversely any point of 1 P will be the dual of a half-space containing P but disjoint with P' and so must belong to &. It follows that 1 P is identical with &.We have therefore established a one-to-one correspondence between the partitions of the Af points and the regions defined by (N-1) half-spaces in E. It follows that

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