Uniform convexity, normal structure and the fixed point property of metric spaces
A metric space \(X\) is said to have the fixed point property (FPP) if the existence of a bounded orbit under a group which acts by isometries on \(X\) implies the presence of a fixed point. The paper introduces a definition of uniform convexity that is applicable beyond geodesic metric spaces. It guarantees a complete metric space satisfies the FPP, as the circumcenter (known to be unique) of the bounded orbit is fixed. The same result is proved for a more general uniform convexity, called \textit{uniform convexity in the wider sense}, which generalizes the related notion in Banach spaces to metric ones. Both uniform convexity and uniform convexity in the wider sense imply compactness and normal structure in the family of admissible subsets of \(X\), properties which are shown to secure the FPP. The author also weakens uniform convexity to the notion of \textit{uniform pseudo-convexity} and shows it implies the FPP as well.
scientific article; zbMATH DE number 6522022
scientific article