  We introduce the notion of trace convexity for functions and respectively,
for subsets of a compact topological space. This notion generalizes both
classical convexity of vector spaces, as well as Choquet convexity for compact
metric spaces. We provide new notions of trace-convexification for sets and
functions as well as a general version of Krein-Milman theorem. We show that
the class of upper semicontinuous convex-trace functions attaining their
maximum at exactly one Choquet-boundary point is residual and we obtain several
enhanced versions of the maximum principle which generalize both the classical
Bauer's theorem as well as its abstract version in the Choquet theory. We
illustrate our notions and results with concrete examples of three different
types.
