The fixed point property and unbounded sets in CAT(0) spaces

Abstract

In this work we study the fixed point property for nonexpansive self-mappings defined on convex and closed subsets of a CAT(0) space. We will show that a positive answer to this problem is very much linked with the Euclidean geometry of the space while the answer is more likely to be negative if the space is more hyperbolic. As a consequence we extend a very well known result of W.O. Ray on Hilbert spaces.

Averaged alternating reflections in geodesic spaces

“…The proof in any other model space M n k follows similar patterns by using the corresponding law of cosines of each space. The above result also holds in the equivalent infinite dimensional spherical and hyperbolic spaces defined using points in ℓ 2 (see, for instance, H ∞ in [14] and [13,Example 3.4]).…”

Geodesic rays, the “Lion-Man” game, and the fixed point property

“…In fact, this characterization of the FPP in terms of geodesic boundedness holds in the setting of complete CAT(0) spaces that are additionally δ-hyperbolic (see [29,Corollary 3.2]). Other results related to the FPP of unbounded sets in geodesic spaces can be found in [12,28,27].…”

“…Such properties include the presence of upper or lower curvature bounds in the sense of Alexandrov [5,12,24], the Gromov hyperbolicity condition [5], the betwenness property [23], or, for normed spaces, the reflexivity [34]. On the other hand, the existence of different types of curves can be used in the analysis of several problems among which we mention the characterization of the fixed point property either for continuous [20,19,24] or nonexpansive mappings [34,12,27], and some pursuit-evasion games [2,23].Geometric properties of a set stand behind the existence of fixed points for continuous or nonexpansive mappings defined on the set in question. This fact has prompted a very fruitful research direction because, among other reasons, it leads to challenging questions regarding the geometry of Banach spaces (see, e.g., the monographs [3,13]).…”

Hyperbolic spaces and directional contractions

“…Thus, Ray's result fails in complete CAT(0) spaces in general. However, it is known that Ray's result does hold in such spaces if their unbounded convex sets satisfy a quasi-boundedness condition which holds for all reflexive Banach spaces (see the recent paper of Espínola [29,Theorem 3.6]). It is also known that a closed convex subset of a complete CAT(κ) space with κ < 0 has the fixed point property for non-expansive mappings if and only if it is geodesically bounded (Piatek [82]).…”