On the Interior of the Convex Hull of a Euclidean Set

This theorem is similar to the well known result that any point in the convex hull of a set E in an w-dimensional euclidean space lies in the convex hull of some subset of E containing at most n-\-l points [l, 2 ] . 1 In these theorems the set E is an arbitrary set in the space. The convex hull of E, denoted by H(E), is the set product of all convex sets in the space which contain E. A euclidean subspace of dimension n — 1 in an w-dimensional euclidean space will be called a plane. Every plane in an ^-dimensional euclidean space separates its complement in the space into two convex open sets, called open half-spaces, whose closures are convex closed sets, called closed half-spaces. If each of the two open halfspaces bounded by a plane L intersects a given set £ , then L is said to be a separating plane of E ; otherwise L is said to be a nonseparating plane of E. In order to prove our sequence of theorems we shall make use of the following result: A point i is interior to the convex hull of a set E in an ^-dimensional euclidean space if and only if ever}'plane through i is a separating plane of £ [l ]. We prove our sequence of theorems by induction. The proof of Theorem Ai is trivial and will be omitted. Now suppose that Theorem An_i is true for an integer n > 1. We shall show that Theorem An is also true. To this end let i be a point interior to the convex hull of a set E in an w-dimensional euclidean space. We are to demonstrate tha t i is interior to the convex hull of some subset P of E containing at most In points. First we show that i is interior to the convex hull of some finite subset Q of E. Since i is interior to H(E)y it is interior to a simplex