Decomposable compact convex sets and peak sets for function spaces.

Geometric conditions are known under which a closed face of a compact convex set is a peak set with respect to the space of continuous affine (real-valued) functions. The purpose of this note is to give an application of this ``abstract-geometric'' set-up to the problem of finding peak sets (or points) in a compact Hausdorff space with respect to a closed subspace of continuous complex-valued functions. In this fashion we obtain the strong hull criteria of Curtis and Figá-Talamanca and in particular the Bishop peak point theorem for function algebras.