The fixed point property for some uniformly nonoctahedral Banach spaces

Abstract

Roughly speaking, we show that a Banach space X has the fixed point property for nonexpansive mappings whenever X has the WORTH property and the unit sphere of X does not contain a triangle with sides of length larger than 2.

Three-dimensional convexity and the fixed point property for nonexpansive mappings

“…In [8] the author introduces a modulus that scales the three-dimensional convexity of the unit ball: He considers the number…”

“…It is evident thatδ X (ε) ≥ δ X (ε) for all ε ∈ (0,δ(X)) and in consequence ε 0 (X) ≤ ε 0 (X). Moreover this last inequality can be strict, since it was shown in [8] the existence of Banach spaces with ε 0 (X) < 2 and which are not uniformly nonsquare. Therefore, the condition ε 0 (X) < 2 can be seen as a three-dimensional generalization of uniform nonsquareness and it is natural to consider the question Problem 1.…”

“…In [8,3] the authors identify properties which, added to ε 0 (X) < 2, imply the FPP. But, as far as we know, there is no result which guarantees the FPP for Banach spaces X with ε 0 (X) < r for some r ∈ (1, 2].…”

Fixed points, normal structure and slices of Banach spaces

“…In this paper, we first study the fixed point property for nonexpansive mappings of a Banach space and some existing result in [7] is extended. We secondly study the relationship between uniform normal structure and slices and some results in [11] are improved too.…”

On James and Jordan–von Neumann constants and the normal structure coefficient of Banach spaces

“…He was then supported by KIT and partly by the Swedish Natural Science Research Council (NFR). At the second visit in 1997 the work was "almost" finished and the first named author presented the results (Theorems 1, 3 and 4) at the conferences in Japan (Kyoto, Aug. 27, 1997-see published results without proofs in the conference proceedings [24]), Poland (Poznań, Aug. 30, 1998) and on seminars in Sweden (Luleå, Sept. 9, 1998) and Spain (Madrid, Sept. 22, 1999). The paper was completed by examples and finally finished when the second named author visited KIT in January-March 2000.…”