The minimum principle for affine functions and isomorphisms of continuous affine function spaces (Q2285064)

Let \(X\) be a compact convex set. In this interesting paper, the authors extend the classical minimum principle by showing for any affine function \(f\) on \(X\) having the point of continuity property that, if it is non-negative on the extreme points of \(X\), then it is a non-negative function. Let \({\mathcal U}_C(X)\) denote the space of real affine continuous function on \(X\). Extending a result on small bound isometries due to \textit{C.-H. Chu} and \textit{H. B. Cohen} [Pac. J. Math. 155, No. 1, 71--86 (1992; Zbl 0728.46011)] in the metrizable case (also called Amir-Cambern type theorems) on such spaces, it is shown that, if a linear isomorphism \(T\) between these function spaces is such that \(\|T\|\|T^{-1}\|<2\), then the underlying compact convex sets have homeomorphic extreme boundaries, under the assumption that every extreme point is a weak peak point.