Towards the fixed point property for superreflexive spaces

Abstract

A Banach space X is said to have property (S m ) if every metrically convex set A C X which lies on the unit sphere and has diameter not greater than one can be (weakly) separated from zero by a functional. We show that this geometrical condition is closely connected with the fixed point property for nonexpansive mappings in superreflexive spaces.

Neocompact Sets and the Fixed Point Property

“…In particular, we give a sufficient condition for a superreflexive Banach space X to have the fixed point property which generalizes the results from [40].…”

“…Corollary 5.8 [40]. If a superreflexive nonstandard hull E has property S m , then E has the fixed point property.…”

“…In [40], we introduced and studied the so-called S m property for a Banach space E. Let us recall the definition. A Banach space E is said to have the S m property if, for every metrically convex set A ⊂ S E with diam A ≤ 1, there exists F ∈ E such that F x > 0 for every x ∈ A.…”

“…In [40], the S m property was proved for various classes of Banach spaces including separable and strictly convex, as well as uniformly noncreasy spaces (see [38] for the definition). In view of Corollary 5.8, it seems to be an important open question whether all superreflexive spaces have property S m or, more generally, whether the assumptions of Theorem 5.7 are satisfied in every superreflexive Banach space.…”

On the super fixed point property in product spaces

“…The following lemma is a reformulation of known facts, see the arguments in [25,Theorem 4.4], [38,Theorem 3.1], [1, p. 86]. We sketch the proof for the convenience of the reader.…”