Equivalence between the fixed point property and reflexivity

Determine whether the fixed point property for Banach spaces is equivalent to reflexivity, thereby resolving the partially proved equivalence between these two properties.

Background

The paper situates renorming theory within metric fixed point theory, where the fixed point property (FPP) concerns the existence of fixed points for suitable nonexpansive mappings on bounded closed convex subsets of Banach spaces. It identifies the equivalence between the FPP and reflexivity as a major unresolved issue that had only been partially established at the time of writing.

The introduction notes that P.-K. Lin constructed a renorming of ℓ1\ell_1 having the FPP, while Tomás Domínguez showed that every reflexive space can be renormed to have the FPP. These results provide partial progress but do not settle the general equivalence stated as the open problem.

References

In the context of the Metric Fixed Point Theory , one of the great open problems, to date partially proven, is that of the equivalence between the FPP and reflexivity, in 2007 PK Lin proved that there is a renorm of $\ell_1$ with the FPP, while in 2008 Tomás Dominguez proved that every reflexive space can be renormed to have the FPP.

— Completeness of constructible norms  (2609.11782 - Acosta-Portilla et al., 10 Sep 2026) in Section 1, Introduction