Introduction to hyperconvex spaces

By Rafa Esṕınola, Mohamed Amine Khamsi. 2001

103 citations recorded by the source corpus.

The notion of hyperconvexity is due to Aronszajn and Panitchpakdi [1] (1956) who proved that a hyperconvex space is a nonexpansive absolute retract, i.e. it is a non-expansive retract of any metric space in which it is isometrically embedded. The corresponding linear theory is well developed and associated with the names of Gleason, Goodner, Kelley and Nachbin (see for instance [19, 29, 42, 46]). The nonlinear theory is still developing. The recent interest into these spaces goes back to the results of Sine [54] and Soardi [57] who proved independently that fixed point property for nonexpansive mappings holds in bounded hyperconvex spaces. Since then many interesting results have been shown to hold in hyperconvex spaces.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/847e1925ce1a73f0c369fe7ef48a6da3975ea893

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Citations in the corpus

Proximinal Retracts and Best Proximity Pair Theorems (2003, background): “For a historical discussion and a detailed survey of the general properties of hyperconvex spaces we refer to Espinola and Khamsi ( 2001 ).”
Some recent results in metric fixed point theory (2007, background): “A discussion of the preceding facts and many more can be found in the survey [ 16 ].”
Invariant approximations for commuting mappings in CAT(0) and hyperconvex spaces (2008, unknown): “, [7–15].”
On asymptotic pointwise contractions in metric spaces (2009, background): “It is well known [2, 7] that if M is hyperconvex, then A(M) is compact and uniformly normal with”
Hyperconvexity and tight-span theory for diversities (2010, methodology): “Hyperconvex metric spaces were deﬁned by Aronszajn and Panitchpakdi in [1] as part of a program to generalise the Hahn-Banach theorem to more general metric spaces (reviewed in [2], and below).”