The minimum principle for affine functions with the point of continuity property and isomorphisms of spaces of continuous affine function

TL;DR

This paper proves that affine functions with the point of continuity property satisfy the minimum principle on compact convex sets and explores isomorphisms of spaces of continuous affine functions, generalizing previous theorems.

Contribution

It establishes the minimum principle for affine functions with the point of continuity property and characterizes isomorphisms of spaces of continuous affine functions under certain conditions.

Findings

Affine functions with the point of continuity property satisfy the minimum principle.

A generalization of Cohen and Chu's theorem relating extreme points and Banach-Mazur distance.

Homeomorphism of extreme point sets under specific isomorphism conditions.

Abstract

Let X be a compact convex set and let ext X stand for the set of extreme points of X. We show that an affine function with the point of continuity property on X satisfies the minimum principle. As a corollary we obtain a generalization of a theorem by H.B. Cohen and C.H. Chu 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 the Banach-Mazur distance between spaces of affine continuous functions on X and Y is smaller then 2. Then ext X is homeomorphic to ext Y.