Weak Compactness and Fixed Point Property for Affine Bi-Lipschitz Maps

TL;DR

This paper characterizes weak compactness of convex sets in Banach spaces via fixed point properties of affine bi-Lipschitz maps, extending previous results to broader classes of maps and introducing new fixed point notions.

Contribution

It generalizes fixed point characterizations of weak compactness to affine bi-Lipschitz maps and introduces the relaxed $ ext{WG}$-$ ext{FPP}$ concept.

Findings

Weakly non-convergent sequences contain wide-$(s)$ subsequences with convex basic sequences.

Weak compactness is characterized by the $ ext{G}$-$ ext{FPP}$ for affine bi-Lipschitz maps.

A convex set is weakly compact iff it has the $ ext{WG}$-$ ext{FPP}$ for affine 1-Lipschitz maps.

Abstract

In this paper we show that if $(y_n)$ is a seminormalized sequence in a Banach space which does not have any weakly convergent subsequence, then it contains a wide-$(s)$ subsequence $(x_n)$ which admits an equivalent convex basic sequence. This fact is used to characterize weak-compactness of bounded, closed convex sets in terms of the generic fixed point property ($\mathcal{G}$-$FPP$) for the class of affine bi-Lipschitz maps. This result generalizes a theorem by Benavides, Jap\'on Pineda and Prus previously proved for the class of continuous maps. We also introduce a relaxation of this notion ($\mathcal{WG}$-$FPP$) and observe that a closed convex bounded subset of a Banach space is weakly compact iff it has the $\mathcal{WG}$-$FPP$ for affine $1$-Lipschitz maps. Related results are also proved. For example, a complete convex bounded subset $C$ of a Hlcs $X$ is weakly compact iff it has the $\mathcal{G}$-$FPP$ for the class of affine continuous maps $f\colon C\to X$ with weak-approximate fixed point nets.