Compact convex sets in non-locally-convex linear spaces

Abstract

In the theorems about compact convex sets it is usually assumed that the compact convex set is contained in a locally convex linear space. Examples for such theorems are the Schauder-Tychonoff fixed point theorem and the Krein-Milman theorem.The problem whether the Krein-Milman theorem remains true also in non locally convex linear spaces, was solved by J.W.Roberts; he gave examples for absolutely convex compact sets without extreme points ([R 76], [R 77], [Ro 84]).It is, however, still an open problem, whether the Schauder-Tychonoff theorem remains true in non locally convex linear spaces [M 81, Problem 54 of Schauder]. A sufficient condition fora compact convex set to have e.g. the fixed point property or to have extreme points is that it can be affinely embedded in a Hausdorff locally convex linear space.This observation, which was the starting point for [JOT 76] and [R 76], was left out of account in various newer publications about Schauder-Tychonoff's fixed point theorem in non locally convex spaces, e.g. [H 84].