Compact Convex Sets and Boundary Integrals

The importance of convexity arguments in functional analysis has long been realized, but a comprehensive theory of infinite-dimensional convex sets has hardly existed for more than a decade. In fact, the integral representation theorems of Choquet and Bishop -de Leeuw together with the uniqueness theorem of Choquet inaugurated a new epoch in infinite-dimensional convexity. Initially considered curious and tech nically difficult, these theorems attracted many mathematicians, and the proofs were gradually simplified and fitted into a general theory. The results can no longer be considered very "deep" or difficult, but they certainly remain all the more important. Today Choquet Theory provides a unified approach to integral representations in fields as diverse as potential theory, probability, function algebras, operator theory, group representations and ergodic theory. At the same time the new concepts and results have made it possible, and relevant, to ask new questions within the abstract theory itself. Such questions pertain to the interplay between compact convex sets K and their associated spaces A(K) of continuous affine functions; to the duality between faces of K and appropriate ideals of A(K); to dominated extension problems for continuous affine functions on faces; and to direct convex sum decomposition into faces, as well as to integral for mulas generalizing such decompositions. These problems are of geometric interest in their own right, but they are primarily suggested by applica tions, in particular to operator theory and function algebras.

I Representations of Points by Boundary Measures.1. Distinguished Classes of Functions on a Compact Convex Set.2. Weak Integrals, Moments and Barycenters.3. Comparison of Measures on a Compact Convex Set.4. Choquet's Theorem.5. Abstract Boundaries Defined by Cones of Functions.6. Unilateral Representation Theorems with Application to Simplicial Boundary Measures.- II Structure of Compact Convex Sets.1. Order-unit and Base-norm Spaces.2. Elementary Embedding Theorems.3. Choquet Simplexes.- 4. Bauer Simplexes and the Dirichlet Problem of the Extreme Boundary.5. Order Ideals, Faces, and Parts.6. Split-faces and Facial Topology.7. The Concept of Center for A(K).8. Existence and Uniqueness of Maximal Central Measures Representing Points of an Arbitrary Compact Convex Set.- References.