For a compact convex set K, let A(K) denote the space of real-valued affine continuous functions, equipped with the supremum norm. For a closed subspace \(X \subset A(K)\) we give sufficient conditions, so that the weak \(^*\) closure of the set of extreme points of the dual unit ball has a decomposition in terms of ‘positive’ and ‘negative’ parts. We give several applications of these ideas to convexity and positivity. When K is a Choquet simplex, we show that the dual unit ball of such an X, inherits nice facial structure. We also use this to partly solve the open problem of exhibiting faces that are Choquet simplexes in the dual unit ball of a Banach space.
