  We characterize inclusions of compact noncommutative convex sets with the
property that every continuous affine function on the smaller set can be
extended to a continuous affine function on the larger set with a uniform
bound. As an application of this result, we obtain a simple geometric
characterization of (possibly nonunital) operator systems that are dualizable,
meaning that their dual can be equipped with an operator system structure. We
further establish some permanence properties of dualizability, and provide a
large new class of dualizable operator systems. These results are new even when
specialized to ordinary compact convex sets.
