Directed Banach spaces of affine functions

Abstract

0. Introduction. Let X be a compact convex set and let F be a closed face of X. In this paper we develop a technique which yields sufficient conditions for F to be a peak-face of X (a subset of X where a continuous affine function on X attains its maximum). The theory is based on a duality between certain types of ordered Banach spaces. This duality is an extension of the results of [6] (see also [17]) and relates the directness of an ordered Banach space E to the degree to which the triangle inequality can be reversed on the positive elements of E*. A precise formulation of this is given in ?1. In ?2 we define a compact convex set X to be at an extreme point x if there is a bounded nonnegative affine function f on X such that f(x) =0 and X= conv ({x} u {y E X: f(y) > 1}). If X is at the G, extreme point x then the results of ?1 are applied to show that x is a peak-point of X. Every compact convex set Xhas a natural identification with the positive elements of norm one in A(X)*, where A(X) is the space of continuous affine functions on X. If N is the subspace of A(X)* spanned by the closed face F of X then by making use of the quotient map from A(X)* to A(X)*/N we can extend the definition of conical to the closed face F. This is then used to establish a sufficient condition for F to be a peak-face. This procedure of using the quotient map is used repeatedly throughout and as a by-product yields different (and possibly simpler) proofs of some known results. For example we use this approach (see Proposition 4.2) to reprove a result of Alfsen's [2] concerning the complementary face of a closed face of a Choquet simplex. In ?3 we define a class Y of compact convex sets X for which it turns out that (1) every closed G, face F of X is a peak-face and (2) every continuous affine function on F can be extended to a continuous affine function on X. It is known that Choquet simplexes have these two properties and we show that Y in fact contains the simplexes. In addition it is proved that Y contains the a-polytopes. (These are defined by R. Phelps [19] and he proves that they correspond exactly to the polyhedrons defined by Alfsen [1].) In [19] Phelps also defines the 3-polytopes as the intersection of a simplex S with a closed subspace of A(S)* of finite codimension. In ?4 we show that the P-polytopes are at each extreme point and that those P-polytopes which are