Abstract
Let X be a compact convex set and let ext X stand for the set of extreme points of X. Let f:X→R be a bounded convex function with the point of continuity property. The first main result shows that f≤0 on X provided f≤0 on ext X. As a byproduct of our method we generalize a result of Raja. Next we show that a resolvable convex semi-extremal nonempty set in X intersects ext X. Finally we prove a Phelps maximum principle for abstract affine functions defined on a locally compact topological space.
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