Compact convex sets in non-locally-convex linear spaces. (English) Zbl 0846.46004

Weber, H.

Note Mat. 12, 271-289 (1992).

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. It is, however, still an open problem, whether the Schauder-Tychonoff theorem remains true in non locally convex linear spaces. A sufficient condition for a 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 was left out of account in various newer publications about Schauder-Tychonoff’s fixed point theorem in non locally convex spaces.
In the first section we give a survey and a precisation of results about sets affinely embeddable in locally convex linear spaces. In the second section we examine conditions for compact convex sets introduced in the literature in connection with fixed point theorems in non locally convex linear spaces. In the last section we prove as consequence of Rosenthal’s lemma that in certain Orlicz sequence spaces for some \(q\in [0,1 ]\) every \(q\)-convex bounded closed subset is compact.

MSC:

46A55	Convex sets in topological linear spaces; Choquet theory
47H10	Fixed-point theorems

Keywords:

compact convex sets; Schauder-Tychonoff fixed point theorem; Krein-Milman theorem; non locally convex linear spaces; absolutely convex compact sets without extreme points; fixed point property; Rosenthal’s lemma; Orlicz sequence spaces