Bijections on the set of extreme points in a compact convex set

TL;DR

This paper explores how gauge-reversing bijections on positive affine functions relate to the structure of extreme points in compact convex sets, extending previous characterizations of JB-algebras.

Contribution

It demonstrates that gauge-reversing bijections are fully determined by their induced mappings on the extreme points of the convex set.

Findings

Gauge-reversing bijections are characterized by their action on extreme points.

The structure of $A(K)$ as a JB-algebra is linked to these bijections.

Extreme points play a central role in understanding these transformations.

Abstract

In a recent work, Roelands and Tiersma proved that, for a compact convex set $K$, the space $A(K)$ of all real-valued continuous affine functions on $K$, is a JB-algebra if and only if there is a gauge-reversing bijection on $A_c(K)$, the set of positive real-valued continuous affine functions on $K$. In this paper, we show that every such gauge-reversing bijection on $A_c(K)$ is completely determined by the induced bijection on the set of extreme points of $K$.