Directed Banach spaces of affine functions

Abstract

0. Introduction.Let A' 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 F to the degree to which the triangle inequality can be reversed on the positive elements of F*.A precise formulation of this is given in §1.In §2 we define a compact convex set X to be conical at an extreme point x if there is a bounded nonnegative affine function /on A'such that/(^) = 0 and X= conv ({x} u {y e X :f(y)£ I}).If X is conical at the Gô extreme point x then the results of §1 are applied to show that x is a peak-point of X.Every compact convex set X has 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 A 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 3P of compact convex sets X for which it turns out that (1) every closed Gô 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 SP in fact contains the simplexes.In addition it is proved that a3 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 ß-polytopes as the intersection of a simplex S with a closed subspace of A(S)* of finite codimension.In §4 we show that the ß-polytopes are conical at each extreme point and that those ß-polytopes which are

Locations

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