Hyperconvex spaces revisited

Abstract

In this paper we describe a construction of a large class of hyperconvex metric spaces. In particular, this construction contains well-known examples of hyperconvex spaces such as R 2 with the "river" metric or with the radial one.Further, we investigate linear hyperconvex spaces with extremal points of their unit balls. We prove that only in the case of a plane (and obviously a line) is there a strict connection between the number of extremal points of the unit ball and the hyperconvexity of the space.Some additional properties concerning the notion of hyperconvexity are also investigated.

On fixed point theorems of Leray–Schauder type

“…It is proved in [2] that a metric so defined is hyperconvex. Let F : X → X be continuous and let H : I × X → X be defined by the formula…”

On Topological Properties of Metrics Defined via Generalized “Linking Construction”

“…In the paper [1] the authors described a construction of a large class of hyperconvex metric spaces (for an introduction to the theory of hyperconvex metric spaces we refer the reader to [2] or [3]). This construction is based on using Chebyshev subsets of a normed space endowed with a hyperconvex metric.…”

On some fixed-point theorems for generalized contractions and their perturbations

“…Also, the notion of hyperconvexity has gained some interest for graph theorists, since Kirk proved in [17] that a hyperconvex metric space with unique metric segments is an R-tree. From the point of view of fixed-point theory it is interesting to emphasize that the fixed-point property holds for nonexpansive mappings in bounded hyperconvex spaces (see [25] and [26] and also [5] for complete generality) as well as that many fixed-point results have been shown to hold in hyperconvex spaces (see, e.g., references given in [7] and [8]). Our new results from Section 5 extend some theorems from [11] and [16].…”