The minimum principle for affine functions and isomorphisms of continuous affine function spaces

Abstrakt

Let X be a compact convex set and let ext X stand for the set of extreme points of X. We show that if f : X -> R is an affine function with the point of continuity property such that f <= 0 on ext X, then f <= 0 on X.

As a corollary of this minimum principle, we obtain a generalization of a theorem by C.H. Chu and H.B.

Cohen by proving the following result. Let X, Y be compact convex sets such that every extreme point of X and Y is a weak peak point and let T : Ac(X).

Ac(Y) be an isomorphism such that parallel to T parallel to . parallel to T-1 parallel to < 2. Then ext X is homeomorphic to ext Y.

Klíčová slova

• Compact convex set
• Affine function
• Point of continuity property
• Banach-Stone theorem