Kirk, William A. (ed.) et al., Handbook of metric fixed point theory. Dordrecht: Kluwer Academic Publishers. 391-435 (2001).
The chapter under review is one of the first such wide treatments of the theory of hyperconvex spaces, with special emphasis put on fixed point theorems. Quite an extensive list of references enables further study of topics not covered in detail.
The first section begins with a short historical introduction. It can also be viewed as a guide for the rest of the chapter.
A proof of the classical Hahn-Banach theorem is the starting point of the second section. The proof is conducted in a way that emphasizes the role of hyperconvexity of the real line; this way, the definition of hyperconvexity which follows seems more natural, as a generalization of a well-known property of the reals.
In the third section, the authors give some properties of hyperconvex spaces; they also define some auxiliary geometric notions connected with subsets of metric spaces and prove their properties in the case of hyperconvex spaces.
The (historically) first result on hyperconvex spaces – the metric counterpart of the Hahn-Banach theorem – is the subject of the next section. A simplified version of Aronszajn and Panitchpakdi’s formulation (see [N. Aronszajn and P. Panitchpakdi, Pac. J. Math. 6, 405-439 (1956; Zbl 0074.17802)]) is proved, so that the reader can learn the idea of the proof without having to pay attention to technicalities.
The fifth section starts with Baillon’s theorem on nonemptiness and hyperconvexity of the intersection of a chain of bounded hyperconvex sets. A notion of a hyperconvex hull is then introduced and two of its important properties are proved: uniqueness (up to an isometry) and preservation of the Hausdorff and Kuratowski measures of noncompactness.
The classical fixed point theorem on nonexpansive mappings of bounded hyperconvex spaces is proved in the sixth section. Two more results on approximate fixed points of nonexpansive maps are proved, one of them not being published elsewhere.
In the seventh section, more fixed point theorems are stated and proved. Here the authors impose compactness type conditions (like being condensing or ultimately compact) on the mappings instead of nonexpansiveness.
The eighth section is completely devoted to Isbell’s construction of a hyperconvex hull of an arbitrary metric space.
In the ninth section, the authors first prove some fixed point and selection theorems for multivalued mappings in hyperconvex spaces. Then results on nonexpansive selections of metric projections follow. These in turn are applied to prove Ky Fan type theorems.
In the tenth section, the Knaster-Kuratowski-Mazurkiewicz principle for hyperconvex spaces is proved. It is then applied to prove a multivalued version of the Ky Fan approximation theorem.
The last, eleventh section deals with one of possible generalizations of hyperconvexity: the so-called “-hyperconvexity”. A generalization of Kirk’s fixed point theorem is then proved.