A convex subset K of a vector space E over the field of real numbers is linearly bounded (linearly closed) if every line intersects K in a bounded (closed) subset of the line. A hyperplane is the set of x ~ E that satisfy a linear equationf(x) = c, wherefis a linear functional and c is a real number.
A main, but not the only, purpose of this note is to establish the following simple theorem, inspired in part by an interesting observation of Karlin's [1], and, in part, by certain anticipated applications.
MAIN THEOREM. Let L be the intersection of a linearly closed and linearly bounded convex set K with n hyperplanes. Then every extreme point of L is a convex combination of at most n 4-1 extreme points of K.
This theorem often simplifies the problem of finding the minimum or maximum of a linear functional restricted to the intersection of a convex set with those vectors that satisfy a given finite number of linear equalities. Such problems, in various guises, arise in many investigations; sometimes the convex set is a set of (probability) measures, sometimes a set of matrices.
Though it costs us something in simplicity and directness of argument, we present the proof as a sequence of lemmas, some of which we label as theorems and corollaries, in order also to make a small contribution to the study of general faces of convex sets.