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

Abstract

Let X be a compact convex set and let ext X stand for the set of extreme points of X. We show that if $f:X\rightarrow {\mathbb {R}}$ is an affine function with the point of continuity property such that $f\le 0$ on ${\text {ext}}\,X$, then $f\le 0$ 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 $T:\mathfrak {A}^c(X)\rightarrow \mathfrak {A}^c(Y)$ be an isomorphism such that $\left\| T\right\| \cdot \left\| T^{-1}\right\| <2$. Then ${\text {ext}}\,X$ is homeomorphic to ${\text {ext}}\,Y$.