A closed convex cone P is said
to be universally well-capped if it contains a compact convex subset B such that
P ∖ B is convex and P = ∪n=1∞nB. The compact convex sets which are universal
caps of some cone are represented as the positive part of the unit ball of an ordered
Banach dual space with the weak∗ topology. A characterization, involving the
directedness of the unit ball, is given of those ordered Banach spaces whose dual
cones are universally well-capped. An application is made to the Choquet
boundary theory for subspaces of continuous functions on a compact Hausdorff
space.
